4: Membrane-Protein Interactions
Membrane proteins are common proteins that are part of, or interact with, biological membranes. Membrane proteins fall into several broad categories depending on their location. Integral membrane proteins are a permanent part of a cell membrane and can either penetrate the membrane (transmembrane) or associate with one or the other side of a membrane (integral monotopic). Peripheral membrane proteins are transiently associated with the cell membrane. Membrane proteins are often medically important—about a third of all human proteins are membrane proteins, and these are targets for more than half of all drugs. Nonetheless, compared to other classes of proteins, determining membrane protein structures remains a challenge in large part due to the difficulty in establishing experimental conditions that can preserve the correct conformation of the protein in isolation from its native environment.
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- 4.1: Membrane Permeability
- All cells are contained by a cell membrane (biomembrane) selectively open to some chemicals and ions but acts as a barrier to undesired components. Here the focus would be on biological membranes in the form of cell membranes, often consist of a phospholipid bilayer with embedded, integral, and/or peripheral proteins responsible for communication and transportation of chemicals and ions.
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- 4.2: Insertion of Membrane Proteins into Lipid Membranes
- Integral membrane proteins are ubiquitous throughout living organisms, ranging from prokaryotes to mammals, accounting for approximately 20-30% of all proteins. They perform a diverse set of functions ranging from signal transduction, to ion transport or even photosynthetic reaction centers. While their activity might vary dramatically, all these proteins experience a similar challenge. They must traverse the amphiphilic lipid membrane to reach their correctly folded state. The ways in which nat
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- 4.3: Protein-lipid Interactions
- lipids also have integral roles such as storing energy or being a major component of a membrane. Despite their individual importance, interaction of lipids and proteins can provide functions that would not be possible individually. The greatest number of these interactions are seen in membranes, which are composed of a wide variety of lipids and proteins.
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- 4.4: Physical Lipid Protein Interactions
- In 1972, Singer and Nicolson proposed the fluid mosaic model which describes the lipids as a neutral environment to the activities of proteins. However, recent studies suggest that lipids are strongly coupled to membrane-proteins and at times the lipid bilayer has the leading role determining the function of membrane-proteins. One specific case of study highlighting the physical lipid-protein interactions concerns the mechanosensitive (MS) ion channels.
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- 4.5: Nanoparticle Spontaneous Penetration and Assembly in and Through Membranes
- Nanomaterial science is a rapidly evolving field of study to approach many scientific questions across fields. From sensors, drug delivery systems, cellular augmentation, and probes to highlight a handful of uses. However, these particles across application often come into contact with lipid membranes and can interact with the berries in a verity of ways. Nanoparticles (NPs) have been shown to restructure lipid membranes to penetrate or embed themselves into the lipids spontaneously. How they ar
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- 4.6: Non-Membrane Lipid Assemblies (Micelles)
- Amphipathic compounds (detergents) are a unique set of molecules with the ability of manipulation (distortion or formation) of the hydrophobic-hydrophilic interactions in biological samples. Detergents in an aqueous solution self-associate to colloid particles. The formation of the colloidal aggregates by detergents are termed micelles. In research micelle-forming detergents provide an amphipathic environment that can mimic lipid bilayers.


12.4: Counting elements of finite sets with bijections
In a future chapter, we will begin learning how to count complicated collections by counting the “choices” needed to determine an arbitrary element in the collection. In this section, we look at how to count collections by finding a collection of the same size which is easier to count.
Consider paths in the \(5 \times 10\) grid below that start at the bottom left and end at the top right, and only move up or right at each step. (One such path is drawn in.) How many such paths are there?
Let \(P\) represent the set of all such paths. We can distinguish each element of \(P\) by the sequence of directions it takes at each step. Let \(\Sigma = \{R,U\}\text{,}\) where \(R\) and \(U\) stand for the directions “right” and “up”, respectively. Then for each path \(p \in P\) we can build a word \(w_p \in \Sigma ^{\ast}\) by setting the letters of \(w_p\) to correspond to the steps in the path. For example, the path in the diagram above would correspond to the word \(RRURUURRRRURR\text{.}\)
This assignment of words in \(\Sigma ^{\ast}\) to paths in \(P\) is a function! Let's call it \(f: P \rightarrow \Sigma ^{\ast}\text{,}\) and set \(W = f(P)\text{,}\) the image of \(f\) in \(\Sigma ^{\ast}\text{.}\) This function is clearly one-to-one, as different paths must produce different words of direction indicators. And since every function maps its domain surjectively onto its image, we obtain a bijection \(f: P \rightarrow W\) by restricting the codomain . Therefore, we can count the paths in \(P\) by instead counting the words in \(W\text{!}\)
Since each path in \(P\) takes exactly \(13\) steps, exactly \(4\) of which must be upwards and exactly \(9\) of which must be to the right, we see that \(W\) consists precisely of those words in \(\Sigma ^{\ast}\) that have length \(13\) and contain exactly \(4\) \(U\)s and \(9\) \(R\)s. Once we learn some basic counting techniques later in the course, you will be able to come back to this example to verify that
\begin{equation*} \vert P \vert = \vert W \vert = 715 \text{.} \end{equation*}


2.6: Permeability
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Permeability describes the effect of material in determining the magnetic flux density. All else being equal, magnetic flux density increases in proportion to permeability.
To illustrate the concept, consider that a particle bearing charge \(q\) moving at velocity \({\bf v}\) gives rise to a magnetic flux density:
\[{\bf B}({\bf r}) = \mu ~ \frac{q {\bf v}}{4\pi R^2} \times \hat{\bf R} %\label{eMatB} \]
where \(\hat{\bf R}\) is the unit vector pointing from the charged particle to the field point \({\bf r}\), \(R\) is this distance, and “\(\times\)” is the cross product. Note that \({\bf B}\) increases with charge and speed, which makes sense since moving charge is the source of the magnetic field. Also note that \({\bf B}\) is inversely proportional to \(4\pi R^2\), indicating that \(\left|{\bf B}\right|\) decreases in proportion to the area of a sphere surrounding the charge, also known as the inverse square law . The remaining factor, \(\mu\), is the constant of proportionality that captures the effect of material. We refer to \(\mu\) as the permeability of the material. Since \({\bf B}\) can be expressed in units of Wb/m\(^2\) and the units of \({\bf v}\) are m/s, we see that \(\mu\) must have units of henries per meter (H/m). (To see this, note that 1 H \(\triangleq\) 1 Wb/A.)
Permeability (\(\mu\), H/m) describes the effect of material in determining the magnetic flux density.
In free space, we find that the permeability \(\mu=\mu_0\) where:
\[\mu_0 = 4\pi \times 10^{-7} ~\mbox{H/m} \nonumber \]
It is common practice to describe the permeability of materials in terms of their relative permeability :
\[\mu_r \triangleq \frac{\mu}{\mu_0} \nonumber \]
which gives the permeability relative to the minimum possible value; i.e., that of free space. Relative permeability for a few representative materials is given in Appendix A2.
Note that \(\mu_r\) is approximately 1 for all but a small class of materials. These are known as magnetic materials , and may exhibit values of \(\mu_r\) as large as \(\sim 10^6\). A commonly-encountered category of magnetic materials is ferromagnetic material, of which the best-known example is iron.


5.3: Styrene Maleic Acid Lipid Particles (SMALP) Technology
Styrene maleic acid lipid particles (SMALPs) are disc-shaped lipid assemblies on the nano-scale made from the interaction of membrane bilayers and styrene maleic acid (SMA) copolymer. SMALP technology holds great potential as a biochemical tool as it allows solubilization of membrane bilayers and its embedded constituent into discs without any use of detergents. SMALPs are thought to be superior to other disc-forming techniques eg. nanodisc as they can form directly from the native biological membrane bilayer and preserve the lipids surrounding the protein of interest. Disc-structure can potentially improve the understanding of membrane proteins by enabling biophysical studies eg. short-angle scattering, dynamic light scattering. Since the use of detergents often destabilizes proteins, SMALPs are advantageous to other established membrane systems.
Styrene Maleic Acid (SMA)
SMA copolymer is formed from polymerization of a mixture of styrene and maleic anhydride in various ratios (3:1 and 2:1 being the most common). The anhydride moieties can subsequently be hydrolyzed to maleic acid (Figure \(\PageIndex{2}\)). The alternating hydrophobic residues (styrene) and hydrophilic (maleic acid) is thought to be determining for the SMA properties of membrane solubilization.
SMA can be purchased from a range of vendors hereunder Cray Valley, Polyscope Polymers (Xiran) and Sigma-Aldrich (3:1 SMA, Lipodisq).
SMA lipid particles (SMALPs)
After originally being patented as Lipodisq, a drug delivery tool for hydrophobic pharmaceuticals 1 , SMALPs with integrated membrane proteins were described by Knowles and coworkers in 2009 2 . The group showed SMALP integration of both the alpha-helical (7TM) protein bacteriorhodopsin (bR) and the beta-barrel protein PagB from dimyristoyl PC liposomes. After purification via Ni 2+ -affinity chromatography, structural analysis dynamic light scattering (DLS), transmission electron microscopy (TEM) and circular dichroism (CD) spectroscopy was carried out. TEM showed nanoparticles of approximately 11 nm in diameter for both proteins, which was consistent with DLS data 2 and results from later studies (12 nm by 3–5 , 9 nm by 1 ). Furthermore, CD spectroscopy showed that both proteins maintained structure after integration into SMALPs while the absorbance of retinal in bR slightly shifted. A similar shift is, however, also seen in detergent. The author attributed this shift to the monomeric quaternary structure in SMALPs and detergent compared to the trimeric structure in the native purple membranes 2 .
Table one contains an overview of the literature concerned on this page and the techniques and concepts that is touched upon.
Table 1 Overview of literature
SMALP formation
Sheidelaar and coworkers 5 investigated the kinetics of SMALP formation (2:1 SMA) from large unilamelar vesicles (LUVs) by monitoring the drop in optical density when LUVs solubilize. The results showed that higher temperature, relative to the lipid T m , and higher SMA concentration both increased kinetics of SMALP formation from di-saturated PC LUVs. This trend was consistent with different various chain lengths. Furthermore, the study showed that the size of the discs was similar regardless of chain length. 5 The exact formation of SMALPs are not entirely clear. Scheidelaar and coworkers propose a three-step model where
- SMA binds the membrane surface,
- destabilize the bilayer before
- the formation of the dis.c 5
Lipid saturation and lipid packing: The lipid packing of the bilayer was shown to strongly influence the SMALP formation kinetics. Even though saturated lipids showed faster kinetics at T>T m , LUVs of mono-unsaturated lipids (that are in the fluid phase at room temperature) showed slower kinetics of SMALP formation.
Accordingly, bilayers of PC with two mono-unsaturated fatty acid chains showed slower kinetics than bilayers of PC with only one mono-unsaturated chain (and one saturated chain) whilst bilayers from PC with two di-unsaturated chain showed similar kinetics (where x is the chain length). Thus the extra double bond in the fatty acid chains does not have a large impact compared to going from saturated to unsaturated. All except di-22:1 PC bilayers, formed SMALPs within 10 min at 30°C. The importance of lipid packing was underlined by results showing a negative correlation between lateral membrane pressure and SMA-mediated solubilization via experiments with bilayers of lipid mixtures with different spatial properties.
Salt concentration: The majority of biological membrane lipids have anionic head groups. Electrostatic repulsion between these head groups and the negatively charged (at physiological pH) maleic acid moieties of SMA might occur. Monitoring SMALP formation from LUVs containing the anionic lipid di-C14:0 PG showed that formation was inhibited in these LUVs in the absence of salt. SMALPs did form in LUVs with mixed lipids. The absence of salt is, however, generally inhibitory to SMA insertion into membranes as it shields the charge repulsion 5 . Sodium chloride and pH buffer (Tris-HCl pH 8) is the only other necessary components to form SMALPs except for SMA and a membrane bilayer eg. a vesicle.
SMALP formation from cell membranes
Sheidelaar and coworkers 5 demonstrated SMALP formation from Escherichia coli membrane fragments to be achievable with excess SMA. Furthermore, the group showed that the membranes fragments, the SMALPs and fatty acids/lipids in the aqueous phase all contained similar relative amounts of PG, PC and cholesterol, thus indicating that SMALPs are not selective for specific lipid types in the membrane bilayer.
Swainsbury and coworkers 4 successfully purified the photoreaction center from the bacterium Rhodobacter sphaeroides directly by SMALP formation from the bacterial membrane with subsequent Ni 2+ -affinity chromatography. The study showed that it was possible to purify membrane proteins/complexes in an active state, in its immediate native lipid environment and underlined that it could be used to study the biochemistry and biophysics in the SMA scaffold. This is interesting as potentially can enable elucidation of the role of nearby lipids for protein function as eg. non - annular lipids.
Conclusion
Styrene maleic acid lipid particles efficiently form disc-shaped nanostructures from intact lipid bilayers from both vesicles and cells with only minor dependency on salt concentration. Biophysical studies can be carried out on proteins embedded in SMALPs in order to study these proteins in the native lipid environment. SMALPs can potentially strengthen the understanding of membrane lipids that are essential for membrane protein function.
References
- Orwick, M. C. et al. Detergent-free formation and physicochemical characterization of nanosized lipid-polymer complexes: Lipodisq. Angew. Chemie - Int. Ed. 51, 4653–4657 (2012).
- Knowles, T. J. et al. Membrane proteins solubilized intact in lipid containing nanoparticles bounded by styrene maleic acid copolymer. J. Am. Chem. Soc. 131, 7484–7485 (2009).
- Orwick-Rydmark, M. et al. Detergent-free incorporation of a seven-transmembrane receptor protein into nanosized bilayer lipodisq particles for functional and biophysical studies. Nano Lett. 12, 4687–4692 (2012).
- Swainsbury, D. J. K., Scheidelaar, S., van Grondelle, R., Killian, J. A. & Jones, M. R. Bacterial Reaction Centers Purified with Styrene Maleic Acid Copolymer Retain Native Membrane Functional Properties and Display Enhanced Stability. Angew. Chemie Int. Ed. 53, 11803–11807 (2014).
- Scheidelaar, S. et al. Molecular Model for the Solubilization of Membranes into Nanodisks by Styrene Maleic Acid Copolymers. Biophys. J. 108, 279–290 (2015).


30.1: Identity Politics in a Fractured Society
The political divisions that plagued the United States in the 1960s were reflected in the rise of identity politics in the 1970s. As people lost hope of reuniting as a society with common interests and goals, many focused on issues of significance to the subgroups to which they belonged, based on culture, ethnicity, sexual orientation, gender, and religion.
HIPPIES AND THE COUNTERCULTURE
In the late 1960s and early 1970s, many young people came to embrace a new wave of cultural dissent. The counterculture offered an alternative to the bland homogeneity of American middle-class life, patriarchal family structures, self-discipline, unquestioning patriotism, and the acquisition of property. In fact, there were many alternative cultures.
Hippies ” rejected the conventions of traditional society. Men sported beards and grew their hair long; both men and women wore clothing from non-Western cultures, defied their parents, rejected social etiquettes and manners, and turned to music as an expression of their sense of self. Casual sex between unmarried men and women was acceptable. Drug use, especially of marijuana and psychedelic drugs like LSD and peyote, was common. Most hippies were also deeply attracted to the ideas of peace and freedom. They protested the war in Vietnam and preached a doctrine of personal freedom to be and act as one wished.
Some hippies dropped out of mainstream society altogether and expressed their disillusionment with the cultural and spiritual limitations of American freedom. They joined communes, usually in rural areas, to share a desire to live closer to nature, respect for the earth, a dislike of modern life, and a disdain for wealth and material goods. Many communes grew their own organic food. Others abolished the concept of private property, and all members shared willingly with one another. Some sought to abolish traditional ideas regarding love and marriage, and free love was practiced openly. One of the most famous communes was The Farm, established in Tennessee in 1971. Residents adopted a blend of Christian and Asian beliefs. They shared housing, owned no private property except tools and clothing, advocated nonviolence, and tried to live as one with nature, becoming vegetarians and avoiding the use of animal products. They smoked marijuana in an effort to reach a higher state of consciousness and to achieve a feeling of oneness and harmony.
Music, especially rock and folk music, occupied an important place in the counterculture. Concerts provided the opportunity to form seemingly impromptu communities to celebrate youth, rebellion, and individuality. In mid-August 1969, nearly 400,000 people attended a music festival in rural Bethel, New York, many for free (Figure \(\PageIndex{2}\)). They jammed roads throughout the state, and thousands had to be turned around and sent home. Thirty-two acts performed for a crowd that partook freely of marijuana, LSD, and alcohol during the rainy three-day event that became known as Woodstock (after the nearby town) and became the cultural touchstone of a generation. No other event better symbolized the cultural independence and freedom of Americans coming of age in the 1960s.
AMERICAN INDIAN PROTEST
As the young, primarily white men and women who became hippies strove to create new identities for themselves, they borrowed liberally from other cultures, including that of Native Americans. At the same time, many Indians were themselves seeking to maintain their culture or retrieve elements that had been lost. In 1968, a group of Indian activists, including Dennis Banks, George Mitchell, and Clyde Bellecourt, convened a gathering of two hundred people in Minneapolis, Minnesota, and formed the American Indian Movement (AIM) (Figure \(\PageIndex{3}\)). The organizers were urban dwellers frustrated by decades of poverty and discrimination. In 1970, the average life expectancy of Indians was forty-six years compared to the national average of sixty-nine. The suicide rate was twice that of the general population, and the infant mortality rate was the highest in the country. Half of all Indians lived on reservations, where unemployment reached 50 percent. Among those in cities, 20 percent lived below the poverty line.
On November 20, 1969, a small group of Indian activists landed on Alcatraz Island (the former site of a notorious federal prison) in San Francisco Bay. They announced plans to build an American Indian cultural center, including a history museum, an ecology center, and a spiritual sanctuary. People on the mainland provided supplies by boat, and celebrities visited Alcatraz to publicize the cause. More people joined the occupiers until, at one point, they numbered about four hundred. From the beginning, the federal government negotiated with them to persuade them to leave. They were reluctant to accede, but over time, the occupiers began to drift away of their own accord. Government forces removed the final holdouts on June 11, 1971, nineteen months after the occupation began.
The next major demonstration came in 1972 when AIM members and others marched on Washington, DC—a journey they called the “Trail of Broken Treaties”—and occupied the offices of the Bureau of Indian Affairs (BIA). The group presented a list of demands, which included improved housing, education, and economic opportunities in Indian communities; the drafting of new treaties; the return of Indian lands; and protections for native religions and culture.
The most dramatic event staged by AIM was the occupation of the Indian community of Wounded Knee, South Dakota, in February 1973. Wounded Knee, on the Pine Ridge Indian Reservation, had historical significance: It was the site of an 1890 massacre of members of the Lakota tribe by the U.S. Army. AIM went to the reservation following the failure of a group of Oglala to impeach the tribal president Dick Wilson, whom they accused of corruption and the use of strong-arm tactics to silence critics. AIM used the occasion to criticize the U.S. government for failing to live up to its treaties with native peoples.
The federal government surrounded the area with U.S. marshals, FBI agents, and other law enforcement forces. A siege ensued that lasted seventy-one days, with frequent gunfire from both sides, wounding a U.S. marshal as well as an FBI agent, and killing two Indians. The government did very little to meet the protesters’ demands. Two AIM leaders, Dennis Banks and Russell Means, were arrested, but charges were later dismissed. The Nixon administration had already halted the federal policy of termination and restored millions of acres to tribes. Increased funding for Indian education, healthcare, legal services, housing, and economic development followed, along with the hiring of more Indian employees in the BIA.
 Many gay rights groups were founded in Los Angeles and San Francisco, cities that were administrative centers in the network of U.S. military installations and the places where many gay men suffered dishonorable discharges. The first postwar organization for homosexual civil rights, the Mattachine Society, was launched in Los Angeles in 1950. The first national organization for lesbians, the Daughters of Bilitis, was founded in San Francisco five years later. In 1966, the city became home to the world’s first organization for transsexual people, the National Transsexual Counseling Unit, and in 1967, the Sexual Freedom League of San Francisco was born.
Through these organizations and others, gay and lesbian activists fought against the criminalization and discrimination of their sexual identities on a number of occasions throughout the 1960s, employing strategies of both protests and litigation. However, the most famous event in the gay rights movement took place not in San Francisco but in New York City. Early in the morning of June 28, 1969, police raided a Greenwich Village gay bar called the Stonewall Inn. Although such raids were common, the response of the Stonewall patrons was anything but. As the police prepared to arrest many of the customers, especially transsexuals and cross-dressers, who were particular targets for police harassment, a crowd began to gather. Angered by the brutal treatment of the prisoners, the crowd attacked. Beer bottles and bricks were thrown. The police barricaded themselves inside the bar and waited for reinforcements. The riot continued for several hours and resumed the following night.
 Gay rights activists protested strongly against the official position of the American Psychiatric Association (APA), which categorized homosexuality as a mental illness and often resulted in job loss, loss of custody, and other serious personal consequences. By 1974, the APA had ceased to classify homosexuality as a form of mental illness but continued to consider it a “sexual orientation disturbance.” Nevertheless, in 1974, Kathy Kozachenko became the first openly lesbian woman voted into office in Ann Arbor, Michigan. In 1977, Harvey Milk became California’s first openly gay man elected to public office, although his service on San Francisco’s board of supervisors, along with that of San Francisco mayor George Moscone, was cut short by the bullet of disgruntled former city supervisor Dan White.
MAYBE NOT NOW
The feminist push for greater rights continued through the 1970s (Figure \(\PageIndex{1}\)). The media often ridiculed feminists as “women’s libbers” and focused on more radical organizations like W.I.T.C.H. (Women’s International Terrorist Conspiracy from Hell), a loose association of activist groups. Many reporters stressed the most unusual goals of the most radical women—calls for the abolition of marriage and demands that manholes be renamed “personholes.”
The majority of feminists, however, sought meaningful accomplishments. In the 1970s, they opened battered women’s shelters and successfully fought for protection from employment discrimination for pregnant women, reform of rape laws (such as the abolition of laws requiring a witness to corroborate a woman’s report of rape), criminalization of domestic violence, and funding for schools that sought to counter sexist stereotypes of women. In 1973, the U.S. Supreme Court in Roe v. Wade affirmed a number of state laws under which abortions obtained during the first three months of pregnancy were legal. This made a nontherapeutic abortion a legal medical procedure nationwide.
Many advances in women’s rights were the result of women’s greater engagement in politics. For example, Patsy Mink, the first Asian American woman elected to Congress, was the co-author of the Education Amendments Act of 1972, Title IX of which prohibits sex discrimination in education. Mink had been interested in fighting discrimination in education since her youth, when she opposed racial segregation in campus housing while a student at the University of Nebraska. She went to law school after being denied admission to medical school because of her gender. Like Mink, many other women sought and won political office, many with the help of the National Women’s Political Caucus (NWPC). In 1971, the NWPC was formed by Bella Abzug, Gloria Steinem, Shirley Chisholm, and other leading feminists to encourage women’s participation in political parties, elect women to office, and raise money for their campaigns (Figure \(\PageIndex{5}\)).\
The ultimate political goal of the National Organization for Women (NOW) was the passage of an Equal Rights Amendment (ERA). The amendment passed Congress in March 1972, and was sent to the states for ratification with a deadline of seven years for passage; if the amendment was not ratified by thirty-eight states by 1979, it would die. Twenty-two states ratified the ERA in 1972, and eight more in 1973. In the next two years, only four states voted for the amendment. In 1979, still four votes short, the amendment received a brief reprieve when Congress agreed to a three-year extension, but it never passed, as the result of the well-organized opposition of Christian and other socially conservative, grassroots organizations.
Section Summary
In the late 1960s and 1970s, Indians, gays and lesbians, and women organized to change discriminatory laws and pursue government support for their interests, a strategy known as identity politics. Others, disenchanted with the status quo, distanced themselves from white, middle-class America by forming their own countercultures centered on a desire for peace, the rejection of material goods and traditional morality, concern for the environment, and drug use in pursuit of spiritual revelations. These groups, whose aims and tactics posed a challenge to the existing state of affairs, often met with hostility from individuals, local officials, and the U.S. government alike. Still, they persisted, determined to further their goals and secure for themselves the rights and privileges to which they were entitled as American citizens.
Review Questions
One of the original founders of AIM was ________.
- Patsy Mink
- Dennis Banks
- Jerry Rubin
- Glenn Weiser
- Answer
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B.
The Supreme Court’s 1973 decision in Roe v. Wade established that ________.
- abortions obtained during the first three months of pregnancy were legal
- witnesses were not required to corroborate a charge of rape
- marriage could not be abolished
- homosexuality was a mental illness
- Answer
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A
What kinds of values did hippies adopt?
- Answer
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Although hippie culture was not entirely homogenous, many hippies desired peace, rejected traditional social values, and sought to live a nonmaterialistic existence close to nature. Many also used drugs both recreationally and as a way to achieve greater spiritual insight.
Glossary
- counterculture
- a culture that develops in opposition to the dominant culture of a society
- identity politics
- political movements or actions intended to further the interests of a particular group membership, based on culture, race, ethnicity, religion, sex, gender, or sexual orientation


20.2: Detecting Antigen-Antibody Complexes in vitro
- Describe various types of assays used to find antigen-antibody complexes
- Describe the circumstances under which antigen-antibody complexes precipitate out of solution
- Explain how antibodies in patient serum can be used to diagnose disease
Laboratory tests to detect antibodies and antigens outside of the body (e.g., in a test tube) are called in vitro assays. When both antibodies and their corresponding antigens are present in a solution, we can often observe a precipitation reaction in which large complexes (lattices) form and settle out of solution. In the next several sections, we will discuss several common in vitro assays.
Precipitin Reactions
A visible antigen-antibody complex is called a precipitin, and in vitro assays that produce a precipitin are called precipitin reactions. A precipitin reaction typically involves adding soluble antigens to a test tube containing a solution of antibodies. Each antibody has two arms, each of which can bind to an epitope. When an antibody binds to two antigens, the two antigens become bound together by the antibody. A lattice can form as antibodies bind more and more antigens together, resulting in a precipitin (Figure \(\PageIndex{1}\)). Most precipitin tests use a polyclonal antiserum rather than monoclonal antibodies because polyclonal antibodies can bind to multiple epitopes, making lattice formation more likely. Although mAbs may bind some antigens, the binding will occur less often, making it much less likely that a visible precipitin will form.
The amount of precipitation also depends on several other factors. For example, precipitation is enhanced when the antibodies have a high affinity for the antigen. While most antibodies bind antigen with high affinity, even high-affinity binding uses relatively weak noncovalent bonds, so that individual interactions will often break and new interactions will occur.
In addition, for precipitin formation to be visible, there must be an optimal ratio of antibody to antigen. The optimal ratio is not likely to be a 1:1 antigen-to-antibody ratio; it can vary dramatically, depending on the number of epitopes on the antigen and the class of antibody. Some antigens may have only one or two epitopes recognized by the antiserum, whereas other antigens may have many different epitopes and/or multiple instances of the same epitope on a single antigen molecule.
Figure \(\PageIndex{2}\) illustrates how the ratio of antigen and antibody affects the amount of precipitation. To achieve the optimal ratio, antigen is slowly added to a solution containing antibodies, and the amount of precipitin is determined qualitatively. Initially, there is not enough antigen to produce visible lattice formation; this is called the zone of antibody excess. As more antigen is added, the reaction enters the equivalence zone (or zone of equivalence), where both the optimal antigen-antibody interaction and maximal precipitation occur. If even more antigen were added, the amount of antigen would become excessive and actually cause the amount of precipitation to decline.
- What is a precipitin?
- Why do polyclonal antisera produce a better precipitin reaction?
Precipitin Ring Test
A variety of techniques allow us to use precipitin formation to quantify either antigen concentration or the amount of antibody present in an antiserum. One such technique is the precipitin ring test (Figure \(\PageIndex{3}\)), which is used to determine the relative amount of antigen-specific antibody in a sample of serum. To perform this test, a set of test tubes is prepared by adding an antigen solution to the bottom of each tube. Each tube receives the same volume of solution, and the concentration of antigens is constant (e.g., 1 mg/mL). Next, glycerol is added to the antigen solution in each test tube, followed by a serial dilution of the antiserum. The glycerol prevents mixing of the antiserum with the antigen solution, allowing antigen-antibody binding to take place only at the interface of the two solutions. The result is a visible ring of precipitin in the tubes that have an antigen-antibody ratio within the equivalence zone. This highest dilution with a visible ring is used to determine the titer of the antibodies. The titer is the reciprocal of the highest dilution showing a positive result, expressed as a whole number. In Figure \(\PageIndex{3}\), the titer is 16.
 In this example, it would not be useful to know what mass of IgG were present in the antiserum, because there are many different specificities of antibody present; but it is important for us to know how much of the antibody activity in a patient’s serum is directed against the antigen of interest (e.g., a particular pathogen or allergen).
Ouchterlony Assay
While the precipitin ring test provides insights into antibody-antigen interactions, it also has some drawbacks. It requires the use of large amounts of serum, and great care must be taken to avoid mixing the solutions and disrupting the ring. Performing a similar test in an agar gel matrix can minimize these problems. This type of assay is variously called double immunodiffusion or the Ouchterlony assay for Orjan Ouchterlony, 1 who first described the technique in 1948.
When agar is highly purified, it produces a clear, colorless gel. Holes are punched in the gel to form wells, and antigen and antisera are added to neighboring wells. Proteins are able to diffuse through the gel, and precipitin arcs form between the wells at the zone of equivalence. Because the precipitin lattice is too large to diffuse through the gel, the arcs are firmly locked in place and easy to see (Figure \(\PageIndex{4}\)).
Although there are now more sensitive and quantitative methods of detecting antibody-antigen interactions, the Ouchterlony test provides a rapid and qualitative way of determining whether an antiserum has antibodies against a particular antigen. The Ouchterlony test is particularly useful when looking for cross-reactivity. We can check an antiserum against a group of closely related antigens and see which combinations form precipitin arcs.
Radial Immunodiffusion Assay
The radial immunodiffusion (RID) assay is similar to the Ouchterlony assay but is used to precisely quantify antigen concentration rather than to compare different antigens. In this assay, the antiserum is added to tempered agar (liquid agar at slightly above 45 °C), which is poured into a small petri dish or onto a glass slide and allowed to cool. Wells are cut in the cooled agar, and antigen is then added to the wells and allowed to diffuse. As the antigen and antibody interact, they form a zone of precipitation. The square of the diameter of the zone of precipitation is directly proportional to the concentration of antigen. By measuring the zones of precipitation produced by samples of known concentration (see the outer ring of samples in Figure \(\PageIndex{5}\)), we can prepare a standard curve for determining the concentration of an unknown solution. The RID assay is a also useful test for determining the concentration of many serum proteins such as the C3 and C4 complement proteins, among others.
- Why does a precipitin ring form in a precipitin ring test, and what are some reasons why a ring might not form?
- Compare and contrast the techniques used in an Ouchterlony assay and a radial immunodiffusion assay.
Flocculation Assays
A flocculation assay is similar to a precipitin reaction except that it involves insoluble antigens such as lipids. A flocculant is similar to a precipitin in that there is a visible lattice of antigen and antibody, but because lipids are insoluble in aqueous solution, they cannot precipitate. Instead of precipitation, flocculation (foaming) is observed in the test tube fluid.
Syphilis is a sexually transmitted infection that can cause severe, chronic disease in adults. In addition, it is readily passed from infected mothers to their newborns during pregnancy and childbirth, often resulting in stillbirth or serious long-term health problems for the infant. Unfortunately, syphilis can also be difficult to diagnose in expectant mothers, because it is often asymptomatic, especially in women. In addition, the causative agent, the bacterium Treponema pallidum , is both difficult to grow on conventional lab media and too small to see using routine microcopy. For these reasons, presumptive diagnoses of syphilis are generally confirmed indirectly in the laboratory using tests that detect antibodies to treponemal antigens.
In 1906, German scientist August von Wassermann (1866–1925) introduced the first test for syphilis that relied on detecting anti-treponemal antibodies in the patient’s blood. The antibodies detected in the Wassermann test were antiphospholipid antibodies that are nonspecific to T. pallidum . Their presence can assist in the diagnosis of syphilis, but because they are nonspecific, they can also lead to false-positive results in patients with other diseases and autoimmune conditions. The original Wasserman test has been modified over the years to minimize false-positives and is now known as the Venereal Disease Research Lab test, better known by its acronym, the VDRL test.
To perform the VDRL test, patient serum or cerebral spinal fluid is placed on a slide with a mixture of cardiolipin (an antigenic phospholipid found in the mitochondrial membrane of various pathogens), lecithin, and cholesterol. The lecithin and cholesterol stabilize the reaction and diminish false positives. Anti-treponemal antibodies from an infected patient’s serum will bind cardiolipin and form a flocculant. Although the VDRL test is more specific than the original Wassermann assay, false positives may still occur in patients with autoimmune diseases that cause extensive cell damage (e.g., systemic lupus erythematosus).
Neutralization Assay
To cause infection, viruses must bind to receptors on host cells. Antiviral antibodies can neutralize viral infections by coating the virions, blocking the binding ( Figure 18.1.6 ). This activity neutralizes virions and can result in the formation of large antibody-virus complexes (which are readily removed by phagocytosis) or by antibody binding to the virus and blocking its binding to host cell receptors. This neutralization activity is the basis of neutralization assays, sensitive assays used for diagnoses of viral infections.
When viruses infect cells, they often cause damage (cytopathic effects) that may include lysis of the host cells. Cytopathic effects can be visualized by growing host cells in a petri dish, covering the cells with a thin layer of agar, and then adding virus (see Isolation, Culture, and Identification of Viruses ). The virus will diffuse very slowly through the agar. A virus will enter a host cell, proliferate (causing cell damage), be released from the dead host cell, and then move to neighboring cells. As more and more cells die, plaques of dead cells will form (Figure \(\PageIndex{6}\)).
During the course of a viral infection, the patient will mount an antibody response to the virus, and we can quantify those antibodies using a plaque reduction assay. To perform the assay, a serial dilution is carried out on a serum sample. Each dilution is then mixed with a standardized amount of the suspect virus. Any virus-specific antibodies in the serum will neutralize some of the virus. The suspensions are then added to host cells in culture to allow any nonneutralized virus to infect the cells and form plaques after several days. The titer is defined as the reciprocal of the highest dilution showing a 50% reduction in plaques. Titer is always expressed as a whole number. For example, if a 1/64 dilution was the highest dilution to show 50% plaque reduction, then the titer is 64.
The presence of antibodies in the patient’s serum does not tell us whether the patient is currently infected or was infected in the past. Current infections can be identified by waiting two weeks and testing another serum sample. A four-fold increase in neutralizing titer in this second sample indicates a new infection.
In a neutralization assay, if a patient’s serum has high numbers of antiviral antibodies, would you expect to see more or fewer plaques?
Immunoelectrophoresis
When a patient has elevated protein levels in the blood or is losing protein in the urine, a clinician will often order a polyacrylamide gel electrophoresis (PAGE) assay (see Visualizing and Characterizing DNA, RNA, and Protein ). This assay compares the relative abundance of the various types of serum proteins. Abnormal protein electrophoresis patterns can be further studied using immunoelectrophoresis (IEP). The IEP begins by running a PAGE. Antisera against selected serum proteins are added to troughs running parallel to the electrophoresis track, forming precipitin arcs similar to those seen in an Ouchterlony assay (Figure \(\PageIndex{7}\)). This allows the identification of abnormal immunoglobulin proteins in the sample.
IEP is particularly useful in the diagnosis of multiple myeloma, a cancer of antibody-secreting cells. Patients with multiple myeloma cannot produce healthy antibodies; instead they produce abnormal antibodies that are monoclonal proteins (M proteins). Thus, patients with multiple myeloma will present with elevated serum protein levels that show a distinct band in the gamma globulin region of a protein electrophoresis gel and a sharp spike (in M protein) on the densitometer scan rather than the normal broad smear (Figure \(\PageIndex{7}\)). When antibodies against the various types of antibody heavy and light chains are used to form precipitin arcs, the M protein will cause distinctly skewed arcs against one class of heavy chain and one class of light chain as seen in Figure \(\PageIndex{7}\).
The advent of electrophoresis ultimately led to researching and understanding the structure of antibodies. When Swedish biochemist Arne Tiselius (1902–1971) published the first protein electrophoresis results in 1937, 2 he could identify the protein albumin (the smallest and most abundant serum protein) by the sharp band it produced in the gel. The other serum proteins could not be resolved in a simple protein electrophoresis, so he named the three broad bands, with many proteins in each band, alpha, beta, and gamma globulins. Two years later, American immunologist Elvin Kabat (1914–2000) traveled to Sweden to work with Tiselius using this new technique and showed that antibodies migrated as gamma globulins. 3 With this new understanding in hand, researchers soon learned that multiple myeloma, because it is a cancer of antibody-secreting cells, could be tentatively diagnosed by the presence of a large M spike in the gamma-globulin region by protein electrophoresis. Prior to this discovery, studies on immunoglobulin structure had been minimal, because of the difficulty of obtaining pure samples to study. Sera from multiple myeloma patients proved to be an excellent source of highly enriched monoclonal immunoglobulin, providing the raw material for studies over the next 20-plus years that resulted in the elucidation of the structure of immunoglobulin.
In general, what does an immunoelectrophoresis assay accomplish?
Immunoblot Assay: The Western Blot
After performing protein gel electrophoresis, specific proteins can be identified in the gel using antibodies. This technique is known as the western blot. Following separation of proteins by PAGE, the protein antigens in the gel are transferred to and immobilized on a nitrocellulose membrane. This membrane can then be exposed to a primary antibody produced to specifically bind to the protein of interest. A second antibody equipped with a molecular beacon will then bind to the first. These secondary antibodies are coupled to another molecule such as an enzyme or a fluorophore (a molecule that fluoresces when excited by light). When using antibodies coupled to enzymes, a chromogenic substrate for the enzyme is added. This substrate is usually colorless but will develop color in the presence of the antibody. The fluorescence or substrate coloring identifies the location of the specific protein in the membrane to which the antibodies are bound (Figure \(\PageIndex{9}\)).
Typically, polyclonal antibodies are used for western blot assays. They are more sensitive than mAbs because of their ability to bind to various epitopes of the primary antigen, and the signal from polyclonal antibodies is typically stronger than that from mAbs. Monoclonal antibodies can also be used; however, they are much more expensive to produce and are less sensitive, since they are only able to recognize one specific epitope.
Several variations of the western blot are useful in research. In a southwestern blot, proteins are separated by SDS-PAGE, blotted onto a nitrocellulose membrane, allowed to renature, and then probed with a fluorescently or radioactively labeled DNA probe; the purpose of the southwestern is to identify specific DNA-protein interactions. Far-western blotsare carried out to determine protein-protein interactions between immobilized proteins (separated by SDS-PAGE, blotted onto a nitrocellulose membrane, and allowed to renature) and non-antibody protein probes. The bound non-antibody proteins that interact with the immobilized proteins in a far-western blot may be detected by radiolabeling, fluorescence, or the use of an antibody with an enzymatic molecular beacon.
What is the function of the enzyme in the immunoblot assay?
Complement-Mediated Immunoassay
One of the key functions of antibodies is the activation (fixation) of complement. When antibody binds to bacteria, for example, certain complement proteins recognize the bound antibody and activate the complement cascade. In response, other complement proteins bind to the bacteria where some serve as opsonins to increase the efficiency of phagocytosis and others create holes in gram-negative bacterial cell membranes, causing lysis. This lytic activity can be used to detect the presence of antibodies against specific antigens in the serum.
Red blood cells are good indicator cells to use when evaluating complement-mediated cytolysis. Hemolysis of red blood cells releases hemoglobin, which is a brightly colored pigment, and hemolysis of even a small number of red cells will cause the solution to become noticeably pink (Figure \(\PageIndex{10}\)). This characteristic plays a role in the complement fixation test, which allows the detection of antibodies against specific pathogens. The complement fixation test can be used to check for antibodies against pathogens that are difficult to culture in the lab such as fungi, viruses, or the bacteria Chlamydia .
To perform the complement fixation test, antigen from a pathogen is added to patient serum. If antibodies to the antigen are present, the antibody will bind the antigen and fix all the available complement. When red blood cells and antibodies against red blood cells are subsequently added to the mix, there will be no complement left to lyse the red cells. Thus, if the solution remains clear, the test is positive. If there are no antipathogen antibodies in the patient’s serum, the added antibodies will activate the complement to lyse the red cells, yielding a negative test (Figure \(\PageIndex{10}\)).
In a complement fixation test, if the serum turns pink, does the patient have antibodies to the antigen or not? Explain.
Table \(\PageIndex{1}\) summarizes the various types of antibody-antigen assays discussed in this section.
| Type of Assay | Mechanism | Examples |
|---|---|---|
| Precipitation | Antibody binds to soluble antigen, forming a visible precipitin | Precipitin ring test to visualize lattice formation in solution |
| Immunoelectrophoresis to examine distribution of antigens following electrophoresis | ||
| Ouchterlony assay to compare diverse antigens | ||
| Radial immunodiffusion assay to quantify antigens | ||
| Flocculation | Antibody binds to insoluble molecules in suspension, forming visible aggregates | VDRL test for syphilis |
| Neutralization | Antibody binds to virus, blocking viral entry into target cells and preventing formation of plaques | Plaque reduction assay for detecting presence of neutralizing antibodies in patient sera |
| Complement activation | Antibody binds to antigen, inducing complement activation and leaving no complement to lyse red blood cells | Complement fixation test for patient antibodies against hard-to-culture bacteria such as Chlamydia |
Key Concepts and Summary
- When present in the correct ratio, antibody and antigen will form a precipitin , or lattice that precipitates out of solution.
- A precipitin ring test can be used to visualize lattice formation in solution. The Ouchterlony assay demonstrates lattice formation in a gel. The radial immunodiffusion assay is used to quantify antigen by measuring the size of a precipitation zone in a gel infused with antibodies.
- Insoluble antigens in suspension will form flocculants when bound by antibodies. This is the basis of the VDRL test for syphilis in which anti-treponemal antibodies bind to cardiolipin in suspension.
- Viral infections can be detected by quantifying virus-neutralizing antibodies in a patient’s serum.
- Different antibody classes in plasma or serum are identified by using immunoelectrophoresis .
- The presence of specific antigens (e.g., bacterial or viral proteins) in serum can be demonstrated by western blot assays, in which the proteins are transferred to a nitrocellulose membrane and identified using labeled antibodies.
- In the complement fixation test, complement is used to detect antibodies against various pathogens.
Footnotes
- 1 Ouchterlony, Örjan, “In Vitro Method for Testing the Toxin-Producing Capacity of Diphtheria Bacteria,” Acta Pathologica Microbiologica Scandinavica 26, no. 4 (1949): 516-24.
- 2 Tiselius, Arne, “Electrophoresis of Serum Globulin: Electrophoretic Analysis of Normal and Immune Sera,” Biochemical Journal 31, no. 9 (1937): 1464.
- 3 Tiselius, Arne and Elvin A. Kabat. “An Electrophoretic Study of Immune Sera and Purified Antibody Preparations,” The Journal of Experimental Medicine 69, no. 1 (1939): 119-31.


5.2: Module 23 Presentation Files
23
Module 23 Presentation Files
Wally Baumback
Learning Outcomes
When you have completed this module, you will be able to:
- Describe a presentation file, an exploded assembly, and an animation.
- Describe and apply the CREATE VIEW, PRECISE VIEW ROTATION, TWEAK COMPONENT, and ANIMATE commands to create, tweak, and play the animation of an exploded assembly.
Presentation Files
After the model is assembled in the assembly file, a presentation file can be created using it. A presentation file is a file in which a view of an assembly is exploded and can be animated. It can also be rotated so that is can be viewed from different angles. A presentation file is created using the NEW command and selecting a presentation template. A presentation file has the extension .ipn. IPN is an acronym for Inventor PreseNtation.
Exploded Views
An exploded view shows the assembly as if it were dismantled and the components of the assembly shown in the order and orientation that they fit together to create the assembly. Assemblies can be exploded automatically by Inventor, tweaked manually, or a combination of the two methods, complete with trails. A tweak is the distance that the part is moved from the grounded part while the trails are the lines in an exploded view that show the relationship of the component to the assembly. Combined, they indicate the direction and distance that a component was moved to create the view. Exploded assembly drawings are used in design presentations, catalogues, and assembly instruction. Figure 23-1 shows an exploded and rotated view of an assembly.

Tweaking
Tweaking is the process of moving the components from the grounded component. The tweak is the distance measured from the grounded unit. In the automated explosion method, the tweak distance is same for each part. If the tweak distance is set to 35, that means the first part is 35 mm from its location and the second part is 70 mm etc. This can be seen if the children in the Browser bar are expanded as shown in Figure 23-2. To tweak components manually, use the TWEAK COMPONENT command.
Animating the Exploded Assembly
After the exploded assembly is tweaked, it can animated to show the assembly and the disassembly of the model. An animation is simply a series of frames or pictures of the assembly displayed one frame at a time. The amount of time between frames is called the interval . The larger the interval, the slower the animation and the shorter the interval, the faster the animation. The number of repetitions can also be set.
Inventor Command: CREATE VIEW or INSERT MODEL
The CREATE VIEW or INSERT MODEL commands are used to insert the model of the assembly file.
Shortcut: none
Inventor Command: TWEAK COMPONENTS
The TWEAK COMPONENTS command is used to move a component farther or closer from the grounded component to create or edit an exploded view of the assembly. It can also be used to rotate the components.
Shortcut: T
Inventor Command: ANIMATE
The ANIMATE command is used to set the parameters and play an animation of an assembly or disassembly of a assembled model.
Shortcut: none
WORK ALONG: Creating an Assembly and Presentation File
Step 1
Check the default project and if necessary, set it to Inventor Course.
Step 2
Use the following instructions to complete all parts in this workalong. Create the following parts and ensure that you do the following:
A
Each part must be saved in its own .ipt file.
B
Project the Center Point onto the Base sketch plane and note the location of X0Y0Z0.
C
Draw the necessary sketches and extrude or revolve them to produce the solid model. Apply all of the necessary geometrical and dimensional constraints to fully constrain it.
D
Apply the colour and material shown.
Step 3
Create the part file as follows:
Part:
Base
Part Name:
Inventor Workalong 23-1A Template: Metric-Modules Part (mm).ipt Color: Aluminum Cast Material: Aluminum 6061 (Figure Step 3A and 3B)
Template: Metric-Modules Part (mm).ipt
Color: Aluminum
Cast Material: Aluminum 6061 (Figure Step 3A and 3B)
Step 4
Change the colour of the machined faces to: Mirror .
Part:
Slotted Wheel
Part Name:
Inventor Workalong 23-1B
Template:
Metric-Modules Part (mm).ipt
Color:
Mirror
Material:
Chrome – Polished Blue (Figure Step 4A, 4B, and 4C)
Part:
Connecting Shaft
Part Name:
Inventor Workalong 23-1C
Template:
Metric-Modules Part (mm).ipt
Color:
Copper – Polished
Material:
Copper (Figure Step 4D and 4E)
Part:
Spacer
Part Name:
Inventor Workalong 23-1D
Template:
Metric-Modules Part (mm).ipt
Color:
Blue – Wall Paint – Glossy
Material:
ABS Plastic (Figure Step 4F and 4G)
Part:
Washer
Part Name:
Inventor Workalong 23-1E
Template:
Metric-Modules Part (mm).ipt
Color:
Steel – Polished
Material:
Steel (Figure Step 4H and 4J)
Part:
Nut
Part Name:
Inventor Workalong 23-1F
Template:
Metric-Modules Part (mm).ipt
Color:
Steel – Polished
Material:
Steel (Figure Step 4K and 4L)
Step 5
Assemble all of the parts that you just created as shown in the figures.
Assembly:
Slotted Connector
Assembly Name:
Inventor Workalong 23-1.iam
Template:
Metric-Modules Assembly (mm).iam (Figure Step 5A and 5B)
Step 6
Enter the NEW command and enable the Metric tab. Select the template file: Modules Presentation (mm).ipn (Figure Step 6)
Step 7
Enter the CREATE VIEW command. It will open the Select Assembly dialogue box. (Figure Step 7)
Step 8
Click the Search Folder icon at the end of the file name. It will open the Open dialogue box. In the Lab Exercises folder, select the file: Inventor Workalong 23-1.iam and click Open. (Figure Step 8)
Step 9
In the Select Assembly dialogue box enable Automatic and Create Trails in the Explosion Method area. Enter the Distance of 35 mm. Click OK. The exploded assembly should appear as shown in the figure. (Figure Step 9A and 9B)
Step 10
Enter the PRECISE VIEW ROTATION command. In the Incremental View Rotate dialogue box, set the Increment to 10. (Figure Step 10)
Step 11
Click the Right Rotate icon twice. This will rotate the model 20 degrees to the right. Click the Up Rotate icon once. This will rotate the model 10 degrees upwards. (Figure Step 11)
Step 12
Enter the TWEAK COMPONENTS command. It will open the Tweak Component dialogue box. Set it as shown in the figure. (Figure Step 12)
Step 13
Ensuring that Direction is enabled, zoom in and move the cursor on the top of the nut. When the direction icon is displayed as shown in the figure, click the left mouse button. (Figure Step 13A and 13B)
Step 14
Look at the Browser bar for the tweak you just created. It will show two tweaks for part:
Inventor Workalong 23-1F:1, which is the nut. The 175.000 mm tweak was created in the automatic explosion in Step 9 and the 185.000 mm from the manual tweak you just completed. (Figure Step 14)
Step 15
Click the Clear button to clear the current settings. Using the Browser bar in Figure Step 15A, manually tweak the assembly to match the figure shown in Figure Step 15B. (Figure Step 15A and 15B)
Step 16
Enter the ANIMATE command. It will open the Animation dialogue box. Set the Interval to 25 and the Repetitions to 1. (Figure Step 16)
Step 17
Study Figure Step 17A, 17B, 17C, and 17D. Play the animation of the assembly that you created in this workalong. Try the different options allowed in the Motion box. (Figure Step 17A, 17B, 17C, and 17D)
Step 18
Set the Interval to 15 and the Repetitions to 3. Play the animation using these parameters both forward and in reverse. (Figure Step 18)
Step 19
Save and close the file.
USER TIP: The tweaked distance can be edited in the Browser bar. Locate the tweak you want to change and select it. Once selected, it will highlight and at the bottom of the browser bar you can change the distance as shown in figure below left. The revised tweak will display as shown in the figure below right.
Key Principles
Key Principles in Module 23
- A presentation file is a file in which a view of an assembly is exploded and can be animated. It can be rotated so that is can be viewed from different angles.
- An exploded view shows the assembly as if it were dismantled and the components of the assembly shown in the order and orientation they fit together to create the assembly.
- A tweak is the distance that the part is moved from the grounded part while the trails are the lines in an exploded view that show the relationship of the component to the assembly.
- After the exploded assembly is tweaked, it can animated to show the assembly and the disassembly of the model. An animation is simply a series of frames or pictures of the assembly displayed one frame at a time. The amount of time between frames is called the interval.


At this time, games for learning (other than management simulations) are being tested in many interesting examples, but it isn’t clear how widely they are being used. Often educators and trainers face obstacles to using computer-based games, including:
-
lack of experience and long learning curves;
-
time and costs required for custom game development and implementation;
-
poor technology support for classroom-based game use; and
-
institutional constraints (learning objective mismatches, standardized fact-based testing, class time constraints, lack of peer acceptance, etc.).
Prensky (2006) notes that curriculum requirements, especially when mandated through legislation, can pose a major obstacle, although “curricular” games are starting to be produced commercially. To overcome this and other obstacles, he suggests the following approaches to beginning to use games in class that ease games into your learning situation without major course restructuring:
-
Bring games played outside class into the classroom through questions, discussions, etc. This can encourage students to reflect on how a game is relevant to a topic and what they are learning from the game. Make game play an assignment for individuals or small groups.
-
Use the principles behind good, complex games to make some or all of your teaching more game-like, and therefore more interesting and engaging to students. One of Prensky’s suggestions for keeping students engaged is to have them vote each time you ask a question.
-
Play a game specifically designed for education in class, such as one of the examples cited above. To do this, you need to become quite familiar with the game in order to handle questions and technical problems.
-
Play a commercial, off-the-shelf game not specifically designed for education, in class, either as a whole class (projected in the front) or as individual students playing separately. Have a student present the game, play the game yourself in front of the class as a springboard to discussion, or divide the students into small groups.


7.3.4: Surface Area of Right Prisms
Lesson
Let's look at the surface area of prisms.
Exercise \(\PageIndex{1}\): Multifaceted
Your teacher will show you a prism.
- What are some things you could measure about the object?
- What units would you use for these measurements?
Exercise \(\PageIndex{2}\): So Many Faces
Here is a picture of your teacher's prism:
Three students are trying to calculate the surface area of this prism.
- Noah says, “This is going to be a lot of work. We have to find the areas of 14 different faces and add them up.”
- Elena says, “It’s not so bad. All 12 rectangles are identical copies, so we can find the area for one of them, multiply that by 12 and then add on the areas of the 2 bases.”
- Andre says, “Wait, I see another way! Imagine unfolding the prism into a net. We can use 1 large rectangle instead of 12 smaller ones.”
- Do you agree with any of them? Explain your reasoning.
- How big is the “1 large rectangle” Andre is talking about? Explain or show your reasoning. If you get stuck, consider drawing a net for the prism.
- Will Noah’s method always work for finding the surface area of any prism? Elena’s method? Andre’s method? Be prepared to explain your reasoning.
- Which method do you prefer? Why?
Exercise \(\PageIndex{3}\): Revisiting a Pentagonal Prism
-
Between you and your partner, choose who will use each of these two methods to find the surface area of the prism.
- Adding the areas of all the faces
- Using the perimeter of the base.
- Use your chosen method to calculate the surface area of the prism. Show your thinking. Organize it so it can be followed by others.
3. Trade papers with your partner, and check their work. Discuss your thinking. If you disagree, work to reach an agreement.
Are you ready for more?
In a deck of cards, each card measures 6 cm by 9 cm.
- When stacked, the deck is 2 cm tall, as shown in the first photo. Find the volume of this deck of cards.
- Then the cards are fanned out, as shown in the second picture. The distance from the rightmost point on the bottom card to the rightmost point on the top card is now 7 cm instead of 2 cm. Find the volume of the new stack.
Summary
To find the surface area of a three-dimensional figure whose faces are made up of polygons, we can find the area of each face, and add them up!
Sometimes there are ways to simplify our work. For example, all of the faces of a cube with side length \(s\) are the same. We can find the area of one face, and multiply by 6. Since the area of one face of a cube is \(s^{2}\), the surface area of a cube is \(6s^{2}\).
We can use this technique to make it faster to find the surface area of any figure that has faces that are the same.
For prisms, there is another way. We can treat the prism as having three parts: two identical bases, and one long rectangle that has been taped along the edges of the bases. The rectangle has the same height as the prism, and its width is the perimeter of the base. To find the surface area, add the area of this rectangle to the areas of the two bases.
Glossary Entries
Definition: Base (of a prism or pyramid)
The word base can also refer to a face of a polyhedron.
A prism has two identical bases that are parallel. A pyramid has one base.
A prism or pyramid is named for the shape of its base.
Definition: Cross Section
A cross section is the new face you see when you slice through a three-dimensional figure.
For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.
Definition: Prism
A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.
Here are some drawings of prisms.
Definition: Pyramid
A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.
Here are some drawings of pyramids.
Definition: Surface Area
The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps.
For example, if the faces of a cube each have an area of 9 cm 2 , then the surface area of the cube is \(6\cdot 9\), or 54 cm 2 .
Definition: Volume
Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps.
For example, the volume of this rectangular prism is 60 units 3 , because it is composed of 3 layers that are each 20 units 3 .
Practice
Exercise \(\PageIndex{4}\)
Edge lengths are given in units. Find the surface area of each prism in square units.
Exercise \(\PageIndex{5}\)
Priya says, “No matter which way you slice this rectangular prism, the cross section will be a rectangle.” Mai says, “I’m not so sure.” Describe a slice that Mai might be thinking of.
(From Unit 7.3.1)
Exercise \(\PageIndex{6}\)
\(B\) is the intersection of line \(AC\) and line \(ED\). Find the measure of each of the angles.
- Angle \(ABF\)
- Angle \(ABD\)
- Angle \(EBC\)
- Angle \(FBC\)
- Angle \(DBG\)
(From Unit 7.1.5)
Exercise \(\PageIndex{7}\)
Write each expression with fewer terms.
- \(12m-4m\)
- \(12m-5k+m\)
- \(9m+k-(3m-2k)\)
(From Unit 6.4.3)
Exercise \(\PageIndex{8}\)
- Find 44% of 625 using the facts that 40% of 625 is 250 and 4% of 625 is 25.
- What is 4.4% of 625?
- What is 0.44% of 625?
(From Unit 4.2.4)


2.3: Time Management Reality
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- Last updated
- Save as PDF
Dave Dillon
“You cannot hoot with the owls and then soar with the eagles.”
– Hubert Humphrey
There is a difference between a goal and a wish. A goal is something that requires action to complete. A wish is something we simply hope will happen without doing anything to achieve it. Students often confuse goals with wishes due to the expected probability of the outcome. For example, a student might say that owning a Ferrari or becoming a movie star were wishes, not goals, because the chance of them happening is slim. We could debate about realistic goals for a long time, but for the purpose of this lesson, the probability of a goal is irrelevant. Think of it like this: the chances of winning the lottery may in fact be slim, but we have no chance to win the lottery if we do not purchase a ticket. Purchasing a ticket requires action, and that distinguishes a difference between a goal and a wish.
When we apply this to education, there are many areas that require action in order to be successful. If I wish for good grades, but spend my time at parties instead of studying, I may not get my wish. But if my goal is to attain good grades, and I take action to achieve them by studying, reviewing, being prepared, etc., then I am much more likely to accomplish my goal.
“Some people want it to happen, some wish it would happen, others make it happen.”
– Michael Jordan
Author’s Story
I had a friend in college who wanted to be a professional athlete. He would talk about it all the time and say that it was his goal. He was a student-athlete but he loved Carl’s Jr. and ate there frequently. He rarely worked out. Over time, I started to think—if he really wanted to be a professional athlete, in order to give himself the best opportunity to make it, he should exercise more and eat a more nutritional diet. It occurred to me years later that he said it was his goal but in reality, it was a wish. He hoped that he would just magically become a professional athlete one day but was unwilling to take the action necessary to help him achieve his goal.
One of the challenges many students face is being over committed. Some are working full-time, going to school full-time, and have other responsibilities as well. Students may additionally be taking care of children, siblings, parents or have other commitments. It can be difficult to take action to complete goals when there are so many areas competing for our time. And sometimes we cannot “do it all.” Sometimes we need to prioritize, let something go, adjust and reevaluate what the most important things are to us.
Other students may struggle because college does not have as much structure as what they may have been used to in high school. Why should I start a homework assignment now when I don’t have anything I have to do for the next three days? This mindset usually leads to the student waiting until the last minute to start the assignment and as a result, the quality of work is not high.
Procrastination
“Do or do not – there is no try.”
– Yoda
Either a homework assignment gets done on time or it doesn’t. Over the years I have seen a lot of excuses from students who didn’t complete their assignment on time.  What level is your internal motivation? How important is the assignment to you? How important is your grade? How important is your class?
Procrastination is the act of putting something off. It’s doing something that’s a low priority instead of doing something that is a high priority. We all procrastinate sometimes. But when we procrastinate on an assignment or studying for an exam until there is little or no time left, our grades suffer and it can be stressful. Learning about why we procrastinate can help us overcome.
Reasons We Procrastinate
I don’t feel like it. I would rather play a video game, watch TV, hang out with friends, sleep, etc. than start my assignment. (The problem is – you might never feel like starting it.)
Perfectionism. I want to do it perfectly and there is not enough time to do it perfectly so I am not going to do it at all.
Fear of success. If I study my tail off and I earn an A on an exam, people will start to expect that I will get A’s all of the time.
Fear of failure. Without confidence, I can’t do the assignment well, no matter how much time or effort I put into it.
“If we are not prepared to fail, we will never create anything original.”
– Sir Ken Robinson
These reasons have been keeping some students from completing assignments and studying for exams. Do you procrastinate? Why?
Whatever the reason may be, procrastination is not a good idea. It often leads to stress. It can be stressful in trying to complete something if we have left it to the last minute. It can be stressful to know that we didn’t submit work that was our best. And stress can take a toll on the health of our bodies.
There are many examples of how American society realizes that people procrastinate. FedEx is built on the fact that people need something immediately and in many cases, they have procrastinated past when regular mail would have gotten it there on time. Post offices stay open later on Tax Day because they know people procrastinated getting their taxes done. Stores offer sales days before Christmas because they know people have procrastinated their Christmas shopping.
So how do we avoid procrastination?
Tell yourself to do your best all of the time. Ask yourself what is important NOW. Other peoples’ expectations of you shouldn’t matter. Be confident in yourself and in your abilities. Do the best you can and be satisfied with your effort. Realize that we’re not perfect. Make your goal to do your best and understand you don’t need to be perfect. Also, realize that you may never “feel like” doing an assignment or studying for an exam.
Get Started. It is the hardest part to do and will have the biggest effect on defeating procrastination. It can be simple: skim the chapter you have to read, think of a title for your paper or schedule an hour for when you will study. The rest of it will be easier once you get started.
Establish and rely on a process. Figure out what works best for you. Take some time to make a plan, list, or outline that allows you to see what you will do and when to complete your assignment or goal. It might be setting aside time early in the morning or waiting to watch a movie until after you’ve finished an assignment. Set your priorities and stick to them.
Set Imaginary Deadlines. If the paper is due in six days, tell yourself it is due in two days. Knock it out early and then enjoy not having it over your head. Fake deadlines are less stressful. And if you do end up needing more time, you have a cushion.
Don’t Break the Chain. Jerry Seinfeld developed a system to help prevent procrastination. He wanted to be a better comedian and believed that writing better jokes would help. To write better jokes, he thought he should write every day. His system, called Don’t Break the Chain was used to motivate himself to write every day. He started with a big wall calendar with a whole year on a page and a red marker. For each day he wrote, he would place a big red X on that day of the calendar. After a few consecutive days, he had a chain. And then the task became not breaking the chain.
Although originally skeptical, I tried it (with exercise), and found it to work. I liked marking the big red X and I liked seeing a long string of big red X’s after a few weeks. I still use this strategy and find myself much more motivated to not break the chain than to go for a run, swim, or to the gym. If there is something you want to practice every day, try it.

– Mark Twain
I don’t suggest that you go out and eat a frog, but the point Twain makes is paramount to overcoming procrastination. He meant if you have to do something you don’t want to, the best thing to do is do it right away: get it over with as soon as possible then move on to enjoy the things you want to do.
This might be true of going to the dentist, making a phone call you don’t want to make, or doing your homework.
Tim Urban’s Ted Talk shines a light on procrastination.
Video: Inside the Mind of a Master Procrastinator, Tim Urban TED Talk
A TED element has been excluded from this version of the text. You can view it online here: https://press.rebus.community/blueprint2/?p=293
Estimating Task Time
One of the biggest challenges I see college students have is accurately estimating how much time it will take to complete a task. We might think we’re going to be able to read an assigned chapter in an hour. But what if it takes three hours to read and understand the chapter? Having the skill to know how long a homework assignment will take is something that can be developed. But until we can anticipate it accurately, it is best to leave some time in our schedule in case it takes longer than we had anticipated.
We have a limited amount of time. Most of us cannot complete everything we wish to complete—either in a day or in a lifetime. We hear people say, “I wish there was more time” or “If there was more time, I would have done this.” We have enough time to do many of the things we wish to do. People run into difficulty when they spend time on things that are not the most important things for them.
“There’s never enough time to do all the nothing you want.”
– Bill Watterson
Author’s Story
I used to say to myself that I would do this or finish that as soon as I got caught up. “Caught up” might apply to my e-mail inbox, keeping current with my twitter timeline, or watching the latest episode of The Walking Dead. But I found that sometimes I was never going to be caught up. So, it was important for me to realize that I was inadvertently placing quadrant III and IV activities ahead of quadrant I and II activities. Worse, I was justifying them by telling myself I would do the I and II activities once the III and IV activities were finished. I corrected this by refocusing on quadrant I and II and constantly reminding myself not to concentrate too much time on the things that are neither urgent nor important.
Time Management Strategies
Laura Vanderkam’s TED Talk helps with perspective on free time.
Video: How to Gain Control of Your Free Time, Laura Vanderkam TED Talk
A TED element has been excluded from this version of the text. You can view it online here: https://press.rebus.community/blueprint2/?p=293
You must make time for the things that are most important to you. In order to make time, you may need to decide you will not do something else.
The ability to say “no” cannot be underestimated. It isn’t easy to say “no,” especially to family, friends and people that like you and whom you like. Most of us don’t want to say “no,” especially when we want to help. But if we always do what others want, we won’t accomplish the things that we want—the things that are most important to us.
Ask yourself:
What am I doing that doesn’t need to be done?
What can I do more efficiently?
Have you ever ordered an appetizer, salad, beverage or bread, then felt full halfway through your entree? In situations like this many people claim, “my eyes were bigger than my stomach.” This is also true with planning and goal setting. It may be that your plan is bigger than the day. Experiment with what you want to accomplish and what is realistic. The better you can accurately predict what you can and will accomplish and how long it will take, the better you can plan, and the more successful you will be.
Licenses and Attributions:
Content previously copyrighted, published in Blueprint for Success in College: Indispensable Study Skills and Time Management Strategies (by Dave Dillon), now licensed as CC BY Attribution.
Tim Urban: Inside the Mind of a Master Procrastinator. Authored by TED.com
Located at:
https://www.ted.com/talks/tim_urban_inside_the_mind_of_a_master_procrastinator
License: CC-BY–NC–ND 4.0 International.
Kim Vanderkam: How to Gain Control of Your Free Time. Authored by TED.com
Located at:
https://www.ted.com/talks/laura_vanderkam_how_to_gain_control_of_your_free_time
License: CC-BY–NC–ND 4.0 International.


6.2: Infinite Sets
Infinite sets are mysterious. Many classical paradoxes address historical confusions about the idea of infinity. At the same time, mathematicians from the ancient Greeks on have found it impossible to develop mathematical thinking without the use of infinity. Why is this so? From a metaphysical point of view, the idea of infinity is probably not necessary. From a physical point of view, there is no evidence for infinity. That is, the universe, as we understand it, is finite. Even from a theological point of view, infinity is to some extent the complement of the finite - and correspondingly gives rise to its own paradoxes.
Infinity has troubled some mathematicians and philosophers, and a few have tried to dispense with it. There aren’t many adherents to this school. The idea of infinity is so useful that the mathematics student will have to develop some comfort with the idea - and its logical consequences. At any rate, infinity clearly exists in the mathematical universe, whether or not it exists in the natural world, and using infinity has been crucial to developing a mathematical understanding of the natural world. In this section we begin an investigation of infinite sets.
We shall use injections and surjections to build some analytical machinery for comparing sets.
NOTATION. \(\preceq\) Let \(X\) and \(Y\) be sets. We write \(X \preceq Y\) if there is an injection \[f: X \rightarrow Y \text {. }\] This notation suggests that, under the conditions of the definition, we think of \(X\) as being "no bigger than" than \(Y\) . This makes sense, since we are able to associate to any element of \(X\) a distinct element of \(Y\) . If \(f\) in the definition is a surjection, then \(f\) is also a bijection and \(|X|=|Y|\) . Otherwise, \(f\) is a function that associates with each element of the range of \(f\) (which is a proper subset of \(Y\) ) a unique element of \(X\) , and \(Y\) still has some elements unaccounted for by \(f\) . So \(Y\) might be "bigger" than \(X\) , but it certainly won’t be "smaller". You might wish to consider this definition in the special case of finite sets \(X\) and \(Y\) . You will observe that \[X \preceq Y \Longleftrightarrow|X| \leq|Y| .\] In Exercise 6.3 you are asked to prove that \(\preceq\) is transitive and reflexive.
REMARK. Are any two sets comparable with respect to \(\preceq\) ? Rather surprisingly, it requires a more advanced assumption, called the Axiom of Choice (see Appendix B), in order to guarantee the comparability of all pairs of sets. Virtually all mathematicians accept the Axiom of Choice. We shall assume the Axiom of Choice in this text.
If \(X \preceq Y\) and \(Y \preceq X\) , we would hope that \(X\) and \(Y\) are the same size. This is indeed true, though the proof is a little tricky. The result is very useful, because it is often much easier to write down two injections than one bijection. THEOREM 6.4. Schröder-Bernstein Theorem Let \(X\) and \(Y\) be sets. If \(X \preceq Y\) and \(Y \preceq X\) , then \(|X|=|Y|\) .
Discussion. The idea behind this proof is as follows. We show that \(|X|=|Y|\) by constructing a bijection \(F: X \nrightarrow Y\) . We are given injections \(f: X \rightarrow Y\) and \(g: Y \rightarrow X\) . We build \(F\) using the injections \(f\) and \(g\) as guides. That is, we wish to define \(F\) so that for each \(x \in X\) , either \(F(x)=f(x)\) or \(F(x)=g^{-1}(x)\) . It is obvious that this cannot be accomplished blindly. For instance, if \(x \in X \backslash g[Y]\) , our hand is forced, and \(F(x)=f(x)\) . Similarly, if \(y \in Y \backslash f[X]\) , our only chance of achieving our objective is for \(F(g(y))=y\) . If we make the wrong choice for \(F(x)\) , we shall lose the use of \(f\) and \(g\) as guides. We might consider \(F\) undecided about \(x\) since \(f\) and \(g^{-1}\) do not agree. The solution is to use \(f\) and \(g\) to move back and forth between \(X\) and \(Y\) until we find that our hand is forced.
Proof. Let \[f: X \rightarrow Y\] and \[g: Y \rightarrow X\] be injections. We may assume that \(X\) and \(Y\) are disjoint.
Discussion. If \(X\) and \(Y\) are not disjoint, we can replace \(X\) with \(X \times\{0\}\) and \(Y\) with \(Y \times\{1\}\) . The existence of a bijection \[g: X \times\{0\} \mapsto Y \times\{1\}\] clearly implies the existence of a bijection from \(X\) to \(Y\) .
If \(x \in X\) we say \(y \in Y\) is the predecessor of \(x\) if \(g(y)=x\) . Analogously, if \(y \in Y\) we say that \(x \in X\) is the predecessor of \(y\) if \(f(x)=y\) . It is possible for an element not to have a predecessor. For example, if \(x \in X \backslash g[Y]\) , then \(x\) has no predecessor. However, if an element does have a predecessor, that predecessor is unique (since \(f\) and \(g\) are both injections). Given an element \(w\) , let \(m(w)\) be 0 if \(w\) does not have a predecessor. Otherwise, let \(m(w)\) be the maximum number \(N \geq 1\) such that there is a finite sequence \(\left\langle z_{n} \mid 0 \leq n \leq N\right\rangle\) for some \(N \geq 1\) satisfying
(1) \(w=z_{N}\)
(2) For \(n<N, z_{n}\) is the predecessor of \(z_{n+1}\) , if the maximum exists. If the maximum doesn’t exist (i.e. if one can make arbitrarily long chains of predecessors), let \(m(w)=\infty\) .
Now define \[\begin{aligned} X_{e} &=\{x \in X \mid m(x) \text { is even }\} \\ X_{o} &=\{x \in X \mid m(x) \text { is odd }\} \\ X_{i} &=\{x \in X \mid m(x)=\infty\} \\ Y_{e} &=\{y \in Y \mid m(y) \text { is even }\} \\ Y_{o} &=\{y \in Y \mid m(y) \text { is odd }\} \\ Y_{i} &=\{y \in Y \mid m(y)=\infty\} \end{aligned}\] The collection \[\left\{X_{e}, X_{o}, X_{i}\right\}\] is obviously a partition of \(X\) . Similarly, \[\left\{Y_{o}, Y_{e}, Y_{i}\right\}\] is a partition of \(Y\) .
We are now in a position to define a bijection between \(X\) and \(Y\) . Let \[F(x)=\left\{\begin{array}{ccc} f(x) & \text { if } & x \in X_{i} \\ f(x) & \text { if } & x \in X_{e} \\ g^{-1}(x) & \text { if } & x \in X_{o} . \end{array}\right.\] DiscussiON. We have some work left in this proof. We need to verify that \(F\) is a bijection from \(X\) to \(Y\) . The idea behind the definition of \(F\) may not be obvious, so let’s investigate the motivation for the definition. Suppose that \(f\) and \(g\) fail to be surjections (if either of the functions is a surjection there would be nothing to prove, since it would also be a bijection). Let \(x \in X \backslash g[Y]\) and \(y \in Y \backslash f[X]\) . Since \(x \notin g[Y]\) , the only possible choice for \(F(x)\) is \(f(x)\) . Similarly, \(y \notin f[X]\) , and the only possible value of \(F^{-1}(y)\) is \(g(y)\) . But this does not solve all of our problems. The set \(X \backslash g[Y]\) is made up of those members of \(X\) that have no predecessors, and \(Y \backslash f[X]\) is composed of the members of \(Y\) with no predecessors. If we are to define \(F\) by piecing together \(f\) and \(g\) , we found that our hands were forced with these sets. Now suppose that \(x \in X\) has exactly one antecedent. Then \(g^{-1}(x)\) has no predecessor. As we observed earlier, we need to satisfy \[F^{-1}\left(g^{-1}(x)\right)=g\left(g^{-1}(x)\right)=x\] and therefore we must satisfy \[F(x)=g^{-1}(x) .\] Similarly, if \(y \in Y\) has exactly one antecedent, we must satisfy \[F^{-1}(y)=f^{-1}(y) .\] If an element \(w\) of \(X \cup Y\) has finitely many antecedents, \(\left.F\right|_{A(w)}\) will be determined by the constraint imposed by the antecedent with no predecessor.
We claim that \[F: X \mapsto Y .\] It is easily seen that \(F\) is well-defined. Since \(X_{o} \subseteq g[Y]\) and \(g\) is an injection, \(\left.F\right|_{X_{o}}=\left.g^{-1}\right|_{X_{o}}\) is well defined. That \(F\) is well defined on \(X_{e}\) and \(X_{i}\) is obvious. Furthermore \[\begin{gathered} F\left[X_{e}\right]=f\left[X_{e}\right]=Y_{o} \\ F\left[X_{o}\right]=g^{-1}\left[X_{o}\right]=Y_{e} \end{gathered}\] and \[F\left[X_{i}\right]=f\left[X_{i}\right]=Y_{i} .\] Discussion. Although we had no choice in the definition of \(F\) on \(X_{e}\) and \(X_{o}\) , we could have defined \(F\) so that \(\left.F\right|_{X_{i}}=\left.g^{-1}\right|_{X_{i}}\) .
Therefore,
\(F[X]=F\left[X_{e} \cup X_{o} \cup X_{i}\right]=f\left[X_{e}\right] \cup g^{-1}\left[X_{o}\right] \cup f\left[X_{i}\right]=Y_{o} \cup Y_{e} \cup Y_{i}=Y .\) So \(F\) is a surjection. We show that \(F\) is an injection. Let \(x, z \in X\) , and suppose \(F(x)=F(z)\) . If \(x \in X_{e}\) , then \(F(x) \in Y_{o}\) and \(z \in X_{e}\) . Hence \[F(x)=f(x)=f(z)=F(z) .\] Since \(f\) is an injection, so is \(\left.f\right|_{X_{e}}\) . Therefore \(x=z\) .
If \(x \in X_{o}\) , then \(F(x) \in Y_{e}\) and \(z \in X_{o}\) . So \[F(x)=g^{-1}(x)=g^{-1}(z)=F(z) .\] The function \(g\) is an injection, therefore \(\left.g^{-1}\right|_{X_{o}}\) is an injection and so \(x=z\) .
Finally, if \(x \in X_{i}\) , then \(F(x) \in X_{i}\) and \(z \in X_{i}\) . So \[F(x)=f(x)=f(z)=F(z) .\] Since \(f\) is an injection, \(x=z\) .
Therefore \(F\) is an injection. Hence, \[F: X \mapsto Y\] and \[|X|=|Y| .\] THEOREM 6.5. \(\mathbb{N}\) is an infinite set.
DiscussiON. We show that any function with domain \(\ulcorner n\urcorner\) , for \(n \in\) \(\mathbb{N}\) , fails to be a surjection. Therefore \(\mathbb{N}\) is not finite.
Proof. Assume \(n \in \mathbb{N}\) , and \[f:\ulcorner n\urcorner \longrightarrow \mathbb{N} .\] Let \[a=1+\sum_{i=0}^{n-1} f(i) \in \mathbb{N} .\] Clearly \(a \notin f[\ulcorner n\urcorner]\) , so \(f\) is not a surjection. Consequently, there is no \(n \in \mathbb{N}\) which can be mapped surjectively onto \(\mathbb{N}\) . Therefore \(\mathbb{N}\) is not finite. Not only is \(\mathbb{N}\) an infinite set, it is in some sense the "smallest" infinite set.
THEOREM 6.6. If \(X\) is infinite, then \(\mathbb{N} \preceq X\) .
Discussion. We shall define an injection \(f: \mathbb{N} \rightarrow X\) inductively, building it up one step at a time.
ProOF. As \(X\) is infinite, it is non-empty, so must contain some element \(x_{0}\) . Define \(f(0)=x_{0}\) .
Now, suppose that \(x_{0}=f(0), x_{1}=f(1), \ldots, x_{n}=f(n)\) have all been chosen, so that \[\left.f\right|_{\{0,1, \ldots, n\}}=\left.f\right|_{n+1\urcorner}: k \mapsto x_{k}\] is injective. As \(X\) is infinite, the function \(\left.f\right|_{\left.{ }_{n+1}\right\urcorner}\) that we have defined cannot be surjective. So there exists some \(x_{n+1}\) in \(X \backslash\left\{x_{0}, \ldots, x_{n}\right\}\) . Define \(f(n+1)=x_{n+1}\) . Continuing in this way, we attain an injection \(f\) defined on all of \(\mathbb{N}\) .
REMARK. The astute reader may have noticed that in the previous proof, we end up making an infinite number of choices of elements of \(X\) .
DEFINITION. Cardinality, \(\aleph_{0}\) We use the expression \(\aleph_{0}\) (read "aleph nought" 1 ) for the size of \(\mathbb{N}\) . That is \[\aleph_{0}:=|\mathbb{N}| \text {. }\] The size of a set is called the cardinality of the set. Any set which is bijective with \(\mathbb{N}\) has cardinality \(\aleph_{0}\) . A finite set has cardinality equal to the unique natural number with which it is bijective.
DEFINITION. Countable A set that is finite or has cardinality \(\aleph_{0}\) is called a countable set.
We are not formally developing the idea of cardinality. This would require working with ordinals, which would distract us from more immediate mathematical interests. However we shall use the language
\({ }^{1} \aleph\) is the first letter of the Hebrew alphabet. and conventions of cardinals where it is intuitive and does not interfere with our program.


12.8: Spotlight on ... Bias in Language and Research
By the end of this section, you will be able to:
- Avoid bias and stereotyping in your writing.
- Evaluate sources for language bias and stereotyping.
When you begin any research project, it is important to have set checks and balances to ensure you are not unknowingly imposing your own biases into your research. A bias is a personal and usually unreasoned judgment, or a prejudice. Although it may be impossible to function without any bias (good or bad), you can consider possible components of bias in your research. No document can be completely objective, for all documents are created by people who have been socialized in some way; therefore, it is helpful to think about the biases of authors of research documents. Many researchers are clear about their biases and state them in introductions to their writings, whereas others may use—or omit—evidence in a way that implies bias for or against the topic.
As a student researcher, seek to be as transparent and critically self-reflective as possible about your preconceptions and language use. Although you will revisit the concept of bias in source selection in The Research Process: Where to Look for Existing Sources, in this section you will look at bias in topic choice and language use.
Language Bias
Language bias ( https://openstax.org/r/languagebias ) refers to words and expressions that are offensive, demeaning, or prejudicial toward individuals or groups on the basis of gender, race, ethnicity, social class, appearance, physical or mental abilities, or sexual orientation. One form of language bias ( https://openstax.org/r/languagebias1 ) is sexist language that includes only one gender. The most common occurrence is use of the word man or men to stand for all people—including those who are not male. Language has changed to be more inclusive, with terms such as firefighter replacing fireman and mail carrier replacing mailman .
Another kind of language bias focuses on people’s health or abilities, indicating a person as a “victim” of a certain disease or “suffering from it.” Use unbiased language to identify the person as someone “with X disease.” Similarly, refer to a “person who is blind” rather than a “blind person.” This kind of language focuses on the person, not the disease or ability.
To avoid language bias, follow these guidelines:
- Use currently accepted terminology when referring to groups of people. If you are writing about a group of people and are unsure of the proper terminology, research the most recent usage patterns before you write. The same is true for pronoun references, about which you can find more information in Editing Focus: Pronouns and Pronouns.
- Be sensitive when referring to people with disabilities by using a “people first” approach. For example, say “a person who uses a wheelchair” instead of “a wheelchair-bound person.”
- Avoid stereotyping—that is, attributing positive or negative characteristics to people on the basis of groups to which they belong. Although not strictly linguistic, stereotyping comes through in making assumptions about people. A stereotypical attitude would be to assume that all or most rich people act in certain ways and, conversely, that all or most poor people act in certain ways. Another stereotypical attitude would be to assume that a person who comes from a particular area of the country adheres to a certain political agenda.
Assignment: Reviewing Your Research Essay for Language Bias
First, review your essay, and speculate as to biases or blocks you may have incorporated while researching and writing. Freewrite, reflecting on these possibilities. Next, go through your essay one section at a time, and highlight any people references, pronouns, biased language, and instances of possible stereotyping. Using the information above and your reflective freewrite, complete a bias chart like the one shown in Table \(12.9\). Next, exchange your essay with two other classmates to see whether they detect bias of which you were unaware. After receiving feedback from two or three people, form a group and discuss what each of you found and how to use alternative language or references. Finally, revisit your freewriting after collaborating with your classmates to reconsider what you wrote before. Add a section to the freewrite in which you relate what your group discussed and what you may have learned from that discussion.
| Essay Section | Essay Language | Possible Bias | Possilbe Revision |
Publish Your Work
After completing your argumentative research paper, you should have a product to be proud of. Your instructor may have a plan to publish papers written by class members, or you may be interested in publishing on your own and joining a wider scholarly conversation. Consider submitting your work to one of these journals that publish undergraduate research:
- Butler Journal of Undergraduate Research ( https://openstax.org/r/butlerjournalUG )
- Crossing Borders: A Multidisciplinary Journal of Undergraduate Scholarship ( https://openstax.org/r/ crossingborders)
- Liberated Arts: A Journal for Undergraduate Research ( https://openstax.org/r/liberatedarts )
- Reinvention: An International Journal of Undergraduate Research ( https://openstax.org/r/reinvention )
In addition, if you have designed your research project as a solution to a local problem, consider finding venues to disseminate the information to those for whom it would be the most useful. This decision may mean adjusting the format, tone, language, and/or conventions of your work to address the needs of a specific audience.