18.5: Spiritual Care of Self
Provision 5 of the American Nurses Association Code of Ethics states, “The nurse owes the same duties to self as to others, including the responsibility to promote health and safety, preserve wholeness of character and integrity, maintain competence, and continue personal and professional growth.” [1] Spiritual care is associated with better health and well-being for everyone, including nurses and nursing students. A desire to help others in need is an important part of spirituality, which has been described as a life-giving force based on faith, discovering meaning and purpose in life, and offering the gift of self to others. [2]
Spiritual resources can help nurses and nursing students overcome the emotional toil associated with caring for seriously ill and dying patients and prevent compassion fatigue and burnout. Read more about compassion fatigue and burnout in the “ Grief and Loss ” chapter.
Many spiritual traditions use contemplative practices to increase compassion, empathy, and quiet the mind. [3] Examples of contemplative practices and other methods to build spiritual strength include the following:
- Meditation can induce feelings of calm and clear-headedness and improve concentration and attention. Research has shown that meditation increases the brain’s gray matter density, which can reduce sensitivity to pain, enhance the immune system, help regulate difficult emotions, and relieve stress. Mindfulness meditation in particular has been proven helpful for people with depression and anxiety, cancer, fibromyalgia, chronic pain, rheumatoid arthritis, type 2 diabetes, chronic fatigue syndrome, and cardiovascular disease. [4]
- Yoga is a centuries-old spiritual practice that creates a sense of union within the practitioner through physical postures, ethical behaviors, and breath expansion. The systematic practice of yoga has been found to reduce inflammation and stress, decrease depression and anxiety, lower blood pressure, and increase feelings of well-being. [5]
- Journaling can help a person become more aware of their inner life and feel more connected to experiences. Studies show that writing during difficult times may help a person find meaning in life’s challenges and become more resilient in the face of obstacles. When journaling, it can be helpful to focus on three basic questions: What experiences give me energy? What experiences drain my energy? Were there any experiences today where I felt alive and experienced “flow”? Allow yourself to write freely, without stopping to edit or worry about spelling and grammar. [6]
- Prayer can elicit the relaxation response, along with feelings of hope, gratitude, and compassion, all of which have a positive effect on overall well-being. There are several types of prayer rooted in the belief that there is a higher power that has some level of influence over one’s life. This belief can provide a sense of comfort and support in difficult times. A recent study found that clinically depressed adults who believed their prayers were heard by a concerned presence responded much better to treatment than those who did not believe. [7]
- Find a spiritual community and friends. Join a spiritual group, such as a church, synagogue, temple, mosque, meditation center, yoga class, or other local group that meets to discuss spiritual issues. The benefits of social support are well-documented, and having a spiritual community to turn to for fellowship can provide a sense of belonging and support. [8]
- Nurture your relationships with family, significant others, and friends. Relationships aren’t static – they are living, dynamic aspects of our lives that require attention and care. To benefit from strong connections with others, you should take charge of your relationships and put in the time and energy you would any other aspect of your well-being. It can be helpful to create rituals together. With busy schedules and the presence of online social media that offer the façade of real contact, it’s very easy to drift from friends. Research has found that people who deliberately make time for gatherings or trips enjoy stronger relationships and more positive energy. An easy way to do this is to create a standing ritual that you can share and that doesn’t create more stress, such as talking on the telephone on Fridays or sharing a walk during lunch breaks. [9]
- Mindfulness has been defined as, “Awareness that arises through paying attention, on purpose, in the present moment, and nonjudgmentally.” Mindfulness has also been described as, “Non-elaborative, nonjudgmental, present-centered awareness in which each thought, feeling, sensation that arises is acknowledged and accepted as it is.” Mindfulness helps us be present in our lives and gives us some control over our reactions and repetitive thought patterns. It helps us pause, get a clearer picture of a situation, and respond more skillfully. Compare your default state to mindfulness when studying for an exam in a difficult course or preparing for a clinical experience. What do you do? Do you tell yourself, “I am not good at this” or “I am going to look stupid”? Does this distract you from paying attention to studying or preparing? How might it be different if you had an open attitude with no concern or judgment about your performance? What if you directly experienced the process as it unfolded, including the challenges, anxieties, insights, and accomplishments, while acknowledging each thought or feeling and accepting it without needing to figure it out or explore it further? If practiced regularly, mindfulness helps a person start to see the habitual patterns that lead to automatic negative reactions that create stress. By observing these thoughts and emotions instead of reacting to them, a person can develop a broader perspective and can choose a more effective response. [10]
- Spending time in nature is cited by many individuals as a spiritual practice that contributes to their mental health. [11]
Explore additional resources about developing spiritual well-being to avoid burnout at the University of Minnesota’s Earl E. Bakken Center for Spirituality & Healing .
- American Nurses Association. (2015). Code of ethics for nurses with interpretive statements. American Nurses Association. https://www.nursingworld.org/practice-policy/nursing-excellence/ethics/code-of-ethics-for-nurses/coe-view-only/ ↵
- Rudolfsson, G., Berggren, I., & da Silva, A. B. (2014). Experiences of spirituality and spiritual values in the context of nursing - An integrative review. The Open Nursing Journal, 8 , 64–70. https://doi.org/10.2174/1874434601408010064 ↵
- Delagran, L. (n.d.). What is spirituality? University of Minnesota. https://www.takingcharge.csh.umn.edu/what-spirituality ↵
- Delagran, L. (n.d.). What is spirituality? University of Minnesota. https://www.takingcharge.csh.umn.edu/what-spirituality ↵
- Delagran, L. (n.d.). What is spirituality? University of Minnesota. https://www.takingcharge.csh.umn.edu/what-spirituality ↵
- Delagran, L. (n.d.). What is spirituality? University of Minnesota. https://www.takingcharge.csh.umn.edu/what-spirituality ↵
- Delagran, L. (n.d.). What is spirituality? University of Minnesota. https://www.takingcharge.csh.umn.edu/what-spirituality ↵
- Delagran, L. (n.d.). What is spirituality? University of Minnesota. https://www.takingcharge.csh.umn.edu/what-spirituality ↵
- Delagran, L. (n.d.). What is spirituality? University of Minnesota. https://www.takingcharge.csh.umn.edu/what-spirituality ↵
- Delagran, L. (n.d.). What is spirituality ? University of Minnesota. https://www.takingcharge.csh.umn.edu/what-spirituality ↵
- Yamada, A., Lukoff, D., Lim, C., & Mancuso, L. (2020). Integrating spirituality and mental health: Perspectives of adults receiving public mental health services in California. Psychology of Religion and Spirituality, 12 (3), 276–287. https://doi.org/10.1037/rel0000260 ↵


7.1: Principle of Mathematical Induction
Axiom \(\PageIndex{1}\)
Suppose \(P(n)\) is a predicate where the variable \(n\) has domain the positive, whole numbers. If
\(P(1)\) is true, and
\((\forall k) (P(k) \rightarrow P(k+1))\) is true,
then \((\forall n)P(n)\) is true.
It is usual to take the principle of mathematical induction as an axiom ; that is, we assume that mathematical induction is valid without proving it.
Below is an outline of the idea behind why it is reasonable to assume that mathematical induction is valid. However, this outline does not constitute a proof since it technically uses mathematical induction implicitly.
Idea
Suppose \(n\) is fixed. We have a sequence of valid arguments:
\(\begin{equation}
\begin{array}
P(1) \rightarrow P(2) & \qquad P(2) \rightarrow P(3) & \qquad \cdots & \qquad P(n-1) \rightarrow P(n) \\
P(1) & \qquad P(2) & & \qquad P(n-1) \\
\hline P(2) & \hline \qquad P(3) && \hline \qquad P(n)
\end{array}
\end{equation}\)
Each is valid (modus ponens). So if we make the two assumptions stated in the principles (i.e. that \(P(1)\) is true and that \(P(k) \rightarrow P(k+1)\) is always true) we can trace the flow of truth from premises to conclusion in each argument in turn:
Definition: First Argument
Premises true so conclusion is true.
Definition: Second Argument
Premises true (using conclusion of the first argument) so conclusion is true.
Definition: Third Argument
…
Definition: \((n-1)^{th}\) argument
Premises true (using conclusion of the \( (n-2)^{th} \) argument) so conclusion is true.
The conclusion of \((n-1)^{th}\) argument is \(P(n)\text{,}\) so \(P(n)\) is true.
Now, here is some specific terminology associated to proofs by induction.
Definition: Base case
the statement \(P(1)\) in a proof by mathematical induction
Definition: Induction Step
the portion of a proof by mathematical induction that establishes the statement \((\forall k) ( P(k) \rightarrow P(k+1) )\)
Definition: Induction Hypothesis
the assumption \(P(k)\) made as the first step in the induction step of a proof by mathematical induction
Procedure \(\PageIndex{1}\): Proof by Induction
To prove a universal statement indexed by whole numbers:
Base case.
Start by proving the statement obtained from the universally quantified predicate for the base case \(n = 1\text{.}\)
Induction step.
Next, assume that \(k\) is a fixed number such that \(k \ge 1\text{,}\) and that the statement obtained from the universally quantified predicate is true for \(n = k\text{.}\) Based on this assumption, try to prove that the next case, \(n=k+1\text{,}\) is also true.
Example \(\PageIndex{1}\)
Prove that the sum of the first \(n\) positive integers is
Solution
We want to prove \((\forall n)P(n)\text{,}\) where \(P(n)\) is as follows.
We will prove this by induction.
Base case.
\(1 = (1\cdot2)/2\) is obviously true.
Induction step.
Assume the statement is true for \(n=k\text{;}\) i.e. assume that
\begin{equation*} 1 + 2 + \cdots + k = k(k+1)/2 \text{.} \end{equation*}
We want to show that this implies the statement is true for \(n = k+1\text{;}\) i.e. show
\begin{equation*} 1 + 2 + \cdots + k + (k+1) = (k+1)(k+2)/2\text{.} \end{equation*}
We have
\begin{align*} 1 + 2 + \cdots + k + (k+1) & = (1 + 2 + \cdots + k) + (k+1) \\ & = \dfrac{k(k+1)}{2} + (k+1) \\ & = \dfrac{k^2 + 3k + 2}{2} \\ & = \dfrac{(k+1)(k+2)}{2} \text{.} \end{align*}
Example \(\PageIndex{2}\)
Prove that \(n^3 + (n+1)^3 + (n+2)^3\) is always divisible by \(9\) for every \(n \ge 1\text{.}\)
Solution
We want to prove \((\forall n)P(n)\text{,}\) where \(P(n)\) is as follows.
\begin{align*} P(1) & \colon \quad 1^3 + 2^3 + 3^3 \text{ is divisible by } 9 \\ P(2) & \colon \quad 2^3 + 3^3 + 4^3 \text{ is divisible by } 9 \\ P(3) & \colon \quad 3^3 + 4^3 + 5^3 \text{ is divisible by } 9 \\ & \vdots \\ P(n) & \colon \quad n^3 + (n+1)^3 + (n+2)^3 \text{ is divisible by } 9 \\ & \vdots \end{align*}
We will prove this by induction.
Base case.
For \(n=1\text{,}\) \(n^3 + (n+1)^3 + (n+2)^3 = 36 = 9\cdot 4\text{.}\)
Induction step.
Assume \(k^3 + (k+1)^3 + (k+2)^3\) is divisible by \(9\text{.}\) This means that there exists some whole number \(m\) so that
\begin{equation*} k^3 + (k+1)^3 + (k+2)^3 = 9 m \text{.} \end{equation*}
We want to show that \((k+1)^3 + (k+2)^3 + (k+3)^3\) is also divisible by \(9\text{.}\) To make the connection between this sum of cubes and the “previous case” sum of cubes above, we can add in (and simultaneously subtract out) a \(k^3\) term:
\begin{align*} (k+1)^3 + (k+2)^3 + (k+3)^3 & = (k^3 + (k+1)^3 + (k+2)^3) + (k+3)^3 - k^3 \\ & = 9m + (k+3)^3 - k^3 \\ & = 9m + (k^3 + 9k^2 + 27k + 27) - k^3 \\ & = 9(m + k^2 + 3k + 3) \text{.} \end{align*}
Since we have factored our sum of cubes into a product involving \(9\text{,}\) that sum of cubes is divisible by \(9\text{.}\)
Example \(\PageIndex{1}\)
Prove that \(3^{3n} + 1\) is divisible by \(7\) whenever \(n\) is odd.
Solution
We do not want to use \(n\) as our induction index, since it jumps by twos. But \(n\) odd means that \(n = 2m - 1\) for some \(m\ge 1\text{,}\) so we want to prove \((\forall m)P(m)\text{,}\) where \(P(m)\) is as follows.
\begin{align*} P(1) & \colon \quad 3^3 + 1 \text{ is divisible by } 7 \\ P(2) & \colon \quad 3^9 + 1 \text{ is divisible by } 7 \\ P(3) & \colon \quad 3^{15} + 1 \text{ is divisible by } 7 \\ & \vdots \\ P(m) & \colon \quad 3^{6m-3} + 1 \text{ is divisible by } 7 \\ & \vdots \end{align*}
We will prove this by induction.
Base case.
For \(m=1\text{,}\) \(3^3+1 = 28 = 7\cdot 4\text{.}\)
Induction step.
Assume \(3^{6k-3} + 1\) is divisible by \(7\text{.}\) This means that there exists some whole number \(\ell\) so that
\begin{equation*} 3^{6k-3} + 1 = 7 \ell \text{.} \end{equation*}
We want to show \(3^{6(k+1)-3} + 1\) is divisible by \(7\text{.}\) We have
\begin{align*} 3^{6 (k + 1) - 3} + 1 & = (3^{6k - 3}) (3^6) + 1 \\ & = (3^{6k-3} + 1 - 1) (3^6) + 1 \\ & = (3^{6k-3} + 1) (3^6) - 3^6 + 1 \\ & = (7 \ell) (3^6) - 728 \\ & = 7 (3^6 \ell - 104) \text{.} \end{align*}
Since we have our expression factored into a product involving \(7\text{,}\) our expression is divisible by \(7\) as desired.
Remark \(\PageIndex{1}\)
Indexing of statements does not have to start at \(1\text{.}\)
Example \(\PageIndex{1}\)
Prove \(2^n \lt n!\) whenever \(n \ge 4\text{.}\)
- Note
-
This statement is actually false for \(n=1,2,3\text{.}\)
Solution
Base case.
For \(n=4\text{,}\) \(2^4 = 16\) and \(4! = 24\text{.}\)
Induction step.
Assume \(2^k \lt k!\) for some \(k \ge 4\text{.}\) We want to show \(2^{k+1} \lt (k+1)!\text{.}\) We have
\begin{equation*} 2^{k+1} = 2(2^k) \lt 2(k!) \lt (k+1)(k!) = (k+1)! \text{.} \end{equation*}


2.10: Strict Four-Voice Composition - Composing in basso continuo Style
Basso continuo (It. for “continuous bass” or “thoroughbass”) is essentially a chordal version of first-species counterpoint. However, instead of composing a single line above a cantus firmus, one composes a succession of chords (performed in the right hand) above a bass line (performed in the left hand). Basso continuo writing, also referred to as realizing a figured bass , gives no consideration to melody, only to the use of proper chords and the smoothest voice-leading possible. Thus, basso continuo style is a simple place to begin engaging the “fundamental musical problems” that arise when more than two lines are combined.
Chord voicing
In strict keyboard-style writing, there are four voices: the bass line (which is usually a given in basso continuo style), and three upper voices : the melody or soprano , the alto , and the tenor (from highest to lowest). Since all three upper voices must be played by a single hand, they should never span more than an octave.
The melody always has an upward-pointing stem. Alto and tenor share a downward-pointing stem. If the alto and tenor share a note, that note receives a single downward-pointing stem. (See m. 1 of the example below.) If melody and alto share a note, that notehead is double-stemmed. (See m. 4 of the example below.)
When choosing the notes to place in the upper voices above a figured bass, use the bass and figures to determine the pitch classes present in the chord. (When realizing an unfigured bass , you must determine appropriate figures before realizing.) If the chord is a four-note chord, use each chord member once, including the bass (exceptions will be noted later). If a chord has three pitch-classes (a triad, for instance), use each pitch-class once, and “double” one of them according to the following principles:
- If the figure is 6/4, 5/3, or other chord of the fifth, double the bass pitch class.
- If the figure is 6/3 and the bass is a fixed scale degree ( do , re , fa , or sol ), double the bass pitch class.
- If the figure is 6/3 and the bass is a variable scale degree ( mi / me , la / le , or ti / te ) or a chromatically altered pitch, double one of the upper voices at the octave or unison.
- Generally, do not double a variable scale degree or a chromatically altered pitch.
In basso continuo style, if the chord is properly voiced (correct pitch classes and correct doublings), two key principles of voice-leading will ensure good counterpoint between the voices most of the time:
- The law of the shortest way (a term coined by composer Arnold Schoenberg): move each voice as little as possible. Prefer repetition to steps, steps to leaps, and one leap at a time to several voices leaping at the same time.
- Move the right hand in contrary or oblique motion to the bass. When the bass leaps by fourth or fifth, though, this rule can be ignored.
In some cases, these rules cannot be followed absolutely (such as when a functional dissonance must be resolved, or when a melody makes it impossible—two cases to be considered later). In all cases, observe the following:
- No parallel fifths or octaves between any pair of voices.
- No contrary fifths or octaves between outer voices.
- Do not approach an octave between the outer voices by similar motion unless the melody moves by step. (All other direct/hidden fifths and octaves are permissible.)


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5.7: Chapter Review - Key Terms
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4.4: Molecular Structure and Polarity
11
Harper College Chemistry Department
Derived from Molecular Structure and Polarity by
Learning Objectives
By the end of this section, you will be able to:
- Predict the structures of small molecules using valence shell electron pair repulsion (VSEPR) theory
- Explain the concepts of polar covalent bonds and molecular polarity
- Assess the polarity of a molecule based on its bonding and structure
Thus far, we have used two-dimensional Lewis structures to represent molecules. However, molecular structure is actually three-dimensional, and it is important to be able to describe molecular bonds in terms of their distances, angles, and relative arrangements in space ( Figure 1 ). A bond angle is the angle between any two bonds that include a common atom, usually measured in degrees. A bond distance (or bond length) is the distance between the nuclei of two bonded atoms along the straight line joining the nuclei. Bond distances are measured in Ångstroms (1 Å = 10 –10 m) or picometers (1 pm = 10 –12 m, 100 pm = 1 Å).
VSEPR Theory
Valence shell electron-pair repulsion theory (VSEPR theory) enables us to predict the molecular structure, including approximate bond angles around a central atom, of a molecule from an examination of the number of bonds and lone electron pairs in its Lewis structure. The VSEPR model assumes that electron pairs in the valence shell of a central atom will adopt an arrangement that minimizes repulsions between these electron pairs by maximizing the distance between them. The electrons in the valence shell of a central atom form either bonding pairs of electrons, located primarily between bonded atoms, or lone pairs. The electrostatic repulsion of these electrons is reduced when the various regions of high electron density assume positions as far from each other as possible.
VSEPR theory predicts the arrangement of electron pairs around each central atom and, usually, the correct arrangement of atoms in a molecule. We should understand, however, that the theory only considers electron-pair repulsions. Other interactions, such as nuclear-nuclear repulsions and nuclear-electron attractions, are also involved in the final arrangement that atoms adopt in a particular molecular structure.
As a simple example of VSEPR theory, let us predict the structure of a gaseous BeF 2 molecule. The Lewis structure of BeF 2 ( Figure 2 ) shows only two electron pairs around the central beryllium atom. With two bonds and no lone pairs of electrons on the central atom, the bonds are as far apart as possible, and the electrostatic repulsion between these regions of high electron density is reduced to a minimum when they are on opposite sides of the central atom. The bond angle is 180° ( Figure 2 ).
Figure 3 illustrates this and other electron-pair geometries that minimize the repulsions among regions of high electron density (bonds and/or lone pairs). Two regions of electron density around a central atom in a molecule form a linear geometry; three regions form a trigonal planar geometry; four regions form a tetrahedral geometry; five regions form a trigonal bipyramidal geometry; and six regions form an octahedral geometry.
Electron-pair Geometry versus Molecular Structure
It is important to note that electron-pair geometry around a central atom is not the same thing as its molecular structure. The electron-pair geometries shown in Figure 3 describe all regions where electrons are located, bonds as well as lone pairs. Molecular structure describes the location of the atoms , not the electrons.
We differentiate between these two situations by naming the geometry that includes all electron pairs the electron-pair geometry . The structure that includes only the placement of the atoms in the molecule is called the molecular structure . The electron-pair geometries will be the same as the molecular structures when there are no lone electron pairs around the central atom, but they will be different when there are lone pairs present on the central atom.
For example, the methane molecule, CH 4 , which is the major component of natural gas, has four bonding pairs of electrons around the central carbon atom; the electron-pair geometry is tetrahedral, as is the molecular structure ( Figure 4 ). On the other hand, the ammonia molecule, NH 3 , also has four electron pairs associated with the nitrogen atom, and thus has a tetrahedral electron-pair geometry. One of these regions, however, is a lone pair, which is not included in the molecular structure, and this lone pair influences the shape of the molecule ( Figure 5 ).
As seen in Figure 5 , small distortions from the ideal angles in Figure 3 can result from differences in repulsion between various regions of electron density. VSEPR theory predicts these distortions by establishing an order of repulsions and an order of the amount of space occupied by different kinds of electron pairs. The order of electron-pair repulsions from greatest to least repulsion is:
This order of repulsions determines the amount of space occupied by different regions of electrons. A lone pair of electrons occupies a larger region of space than the electrons in a triple bond; in turn, electrons in a triple bond occupy more space than those in a double bond, and so on. The order of sizes from largest to smallest is:
Consider formaldehyde, H 2 CO, which is used as a preservative for biological and anatomical specimens ( Figure 1 ). This molecule has regions of high electron density that consist of two single bonds and one double bond. The basic geometry is trigonal planar with 120° bond angles, but we see that the double bond causes slightly larger angles (121°), and the angle between the single bonds is slightly smaller (118°).
In the ammonia molecule, the three hydrogen atoms attached to the central nitrogen are not arranged in a flat, trigonal planar molecular structure, but rather in a three-dimensional trigonal pyramid ( Figure 5 ) with the nitrogen atom at the apex and the three hydrogen atoms forming the base. The ideal bond angles in a trigonal pyramid are based on the tetrahedral electron pair geometry. Again, there are slight deviations from the ideal because lone pairs occupy larger regions of space than do bonding electrons. The H–N–H bond angles in NH 3 are slightly smaller than the 109.5° angle in a regular tetrahedron ( Figure 4 ) because the lone pair-bonding pair repulsion is greater than the bonding pair-bonding pair repulsion ( Figure 5 ). Figure 6 illustrates the ideal molecular structures, which are predicted based on the electron-pair geometries for various combinations of lone pairs and bonding pairs.
According to VSEPR theory, the terminal atom locations (Xs in Figure 6 ) are equivalent within the linear, trigonal planar, and tetrahedral electron-pair geometries (the first three rows of the table). It does not matter which X is replaced with a lone pair because the molecules can be rotated to convert positions.
Predicting Electron Pair Geometry and Molecular Structure
The following procedure uses VSEPR theory to determine the electron pair geometries and the molecular structures:
- Write the Lewis structure of the molecule or polyatomic ion.
- Count the number of regions of electron density (lone pairs and bonds) around the central atom. A single, double, or triple bond counts as one region of electron density.
- Identify the electron-pair geometry based on the number of regions of electron density: linear, trigonal planar, or tetrahedral ( Figure 6 , first column).
- Use the number of lone pairs to determine the molecular structure ( Figure 6 ). If more than one arrangement of lone pairs and chemical bonds is possible, choose the one that will minimize repulsions, remembering that lone pairs occupy more space than multiple bonds, which occupy more space than single bonds. In trigonal bipyramidal arrangements, repulsion is minimized when every lone pair is in an equatorial position. In an octahedral arrangement with two lone pairs, repulsion is minimized when the lone pairs are on opposite sides of the central atom.
The following examples illustrate the use of VSEPR theory to predict the molecular structure of molecules or ions that have no lone pairs of electrons. In this case, the molecular structure is identical to the electron pair geometry.
EXAMPLE 1
Predicting Electron-pair Geometry and Molecular Structure: CO 2 and BCl 3
Predict the electron-pair geometry and molecular structure for each of the following:
(a) carbon dioxide, CO 2 , a molecule produced by the combustion of fossil fuels
(b) boron trichloride, BCl 3 , an important industrial chemical
Solution
(a) We write the Lewis structure of CO 2 as:
This shows us two regions of high electron density around the carbon atom—each double bond counts as one region, and there are no lone pairs on the carbon atom. Using VSEPR theory, we predict that the two regions of electron density arrange themselves on opposite sides of the central atom with a bond angle of 180°. The electron-pair geometry and molecular structure are identical, and CO 2 molecules are linear.
(b) We write the Lewis structure of BCl 3 as:
Thus we see that BCl 3 contains three bonds, and there are no lone pairs of electrons on boron. The arrangement of three regions of high electron density gives a trigonal planar electron-pair geometry. The B–Cl bonds lie in a plane with 120° angles between them. BCl 3 also has a trigonal planar molecular structure ( Figure 7.21 ).
The electron-pair geometry and molecular structure of BCl 3 are both trigonal planar. Note that the VSEPR geometry indicates the correct bond angles (120°), unlike the Lewis structure shown above.
Check Your Learning
Carbonate, “> CO 3 2− , CO32−, is a common polyatomic ion found in various materials from eggshells to antacids. What are the electron-pair geometry and molecular structure of this polyatomic ion?
Answer:
The electron-pair geometry is trigonal planar and the molecular structure is trigonal planar. Due to resonance, all three C–O bonds are identical. Whether they are single, double, or an average of the two, each bond counts as one region of electron density.
EXAMPLE 2
Predicting Electron-pair Geometry and Molecular Structure: Ammonium
Two of the top 50 chemicals produced in the United States, ammonium nitrate and ammonium sulfate, both used as fertilizers, contain the ammonium ion. Predict the electron-pair geometry and molecular structure of the “> NH 4 + NH4+ cation.
Solution
We write the Lewis structure of “> NH 4 + NH4+ as:
We can see that “> NH 4 + NH4+ contains four bonds from the nitrogen atom to hydrogen atoms and no lone pairs. We expect the four regions of high electron density to arrange themselves so that they point to the corners of a tetrahedron with the central nitrogen atom in the middle ( Figure 7.19 ). Therefore, the electron pair geometry of “> NH 4 + NH4+ is tetrahedral, and the molecular structure is also tetrahedral ( Figure 7.22 ).
Check Your Learning
Identify a molecule with trigonal bipyramidal molecular structure.
Answer:
Any molecule with five electron pairs around the central atoms including no lone pairs will be trigonal bipyramidal. PF 5 is a common example.
The next several examples illustrate the effect of lone pairs of electrons on molecular structure.
EXAMPLE 3
Predicting Electron-pair Geometry and Molecular Structure: Lone Pairs on the Central Atom
Predict the electron-pair geometry and molecular structure of a water molecule.
Solution
The Lewis structure of H 2 O indicates that there are four regions of high electron density around the oxygen atom: two lone pairs and two chemical bonds:
We predict that these four regions are arranged in a tetrahedral fashion ( Figure 7.23 ), as indicated in Figure 6 . Thus, the electron-pair geometry is tetrahedral and the molecular structure is bent with an angle slightly less than 109.5°. In fact, the bond angle is 104.5°.
Check Your Learning
The hydronium ion, H 3 O + , forms when acids are dissolved in water. Predict the electron-pair geometry and molecular structure of this cation.
Answer:
electron pair geometry: tetrahedral; molecular structure: trigonal pyramidal
Check Your Learning
In a certain molecule, the central atom has three lone pairs and two bonds. What will the electron pair geometry and molecular structure be?
Answer:
electron pair geometry: trigonal bipyramidal; molecular structure: linear
Molecular Structure for Multicenter Molecules
When a molecule or polyatomic ion has only one central atom, the molecular structure completely describes the shape of the molecule. Larger molecules do not have a single central atom, but are connected by a chain of interior atoms that each possess a “local” geometry. The way these local structures are oriented with respect to each other also influences the molecular shape, but such considerations are largely beyond the scope of this introductory discussion. For our purposes, we will only focus on determining the local structures.
EXAMPLE 7.16
Predicting Structure in Multicenter Molecules
The Lewis structure for the simplest amino acid, glycine, H 2 NCH 2 CO 2 H, is shown here. Predict the local geometry for the nitrogen atom, the two carbon atoms, and the oxygen atom with a hydrogen atom attached:
Solution
Consider each central atom independently. The electron-pair geometries:
- nitrogen–four regions of electron density; tetrahedral
- carbon ( C H 2 )–four regions of electron density; tetrahedral
- carbon ( C O 2 )—three regions of electron density; trigonal planar
- oxygen ( O H)—four regions of electron density; tetrahedral
The local structures:
- nitrogen–three bonds, one lone pair; trigonal pyramidal
- carbon ( C H 2 )—four bonds, no lone pairs; tetrahedral
- carbon ( C O 2 )—three bonds (double bond counts as one bond), no lone pairs; trigonal planar
- oxygen ( O H)—two bonds, two lone pairs; bent (109°)
Check Your Learning
Another amino acid is alanine, which has the Lewis structure shown here. Predict the electron-pair geometry and local structure of the nitrogen atom, the three carbon atoms, and the oxygen atom with hydrogen attached:
Answer:
electron-pair geometries: nitrogen––tetrahedral; carbon ( C H)—tetrahedral; carbon ( C H 3 )—tetrahedral; carbon ( C O 2 )—trigonal planar; oxygen ( O H)—tetrahedral; local structures: nitrogen—trigonal pyramidal; carbon ( C H)—tetrahedral; carbon ( C H 3 )—tetrahedral; carbon ( C O 2 )—trigonal planar; oxygen ( O H)—bent (109°)
LINK TO LEARNING
The molecular shape simulator lets you build various molecules and practice naming their electron-pair geometries and molecular structures.
EXAMPLE 7.17
Molecular Simulation
Using molecular shape simulator allows us to control whether bond angles and/or lone pairs are displayed by checking or unchecking the boxes under “Options” on the right. We can also use the “Name” checkboxes at bottom-left to display or hide the electron pair geometry (called “electron geometry” in the simulator) and/or molecular structure (called “molecular shape” in the simulator).
Build the molecule HCN in the simulator based on the following Lewis structure:
Click on each bond type or lone pair at right to add that group to the central atom. Once you have the complete molecule, rotate it to examine the predicted molecular structure. What molecular structure is this?
Solution
The molecular structure is linear.
Check Your Learning
Build a more complex molecule in the simulator. Identify the electron-group geometry, molecular structure, and bond angles. Then try to find a chemical formula that would match the structure you have drawn.
Answer:
Answers will vary. For example, an atom with four single bonds, a double bond, and a lone pair has an octahedral electron-group geometry and a square pyramidal molecular structure. XeOF 4 is a molecule that adopts this structure.
Molecular Polarity and Dipole Moment
As discussed previously, polar covalent bonds connect two atoms with differing electronegativities, leaving one atom with a partial positive charge (δ+) and the other atom with a partial negative charge (δ–), as the electrons are pulled toward the more electronegative atom. This separation of charge gives rise to a bond dipole moment . The magnitude of a bond dipole moment is represented by the Greek letter mu ( µ ) and is given by the formula shown here, where Q is the magnitude of the partial charges (determined by the electronegativity difference) and r is the distance between the charges:
This bond moment can be represented as a vector , a quantity having both direction and magnitude ( Figure 7.26 ). Dipole vectors are shown as arrows pointing along the bond from the less electronegative atom toward the more electronegative atom. A small plus sign is drawn on the less electronegative end to indicate the partially positive end of the bond. The length of the arrow is proportional to the magnitude of the electronegativity difference between the two atoms.
A whole molecule may also have a separation of charge, depending on its molecular structure and the polarity of each of its bonds. If such a charge separation exists, the molecule is said to be a polar molecule (or dipole); otherwise the molecule is said to be nonpolar. The dipole moment measures the extent of net charge separation in the molecule as a whole. We determine the dipole moment by adding the bond moments in three-dimensional space, taking into account the molecular structure.
For diatomic molecules, there is only one bond, so its bond dipole moment determines the molecular polarity. Homonuclear diatomic molecules such as Br 2 and N 2 have no difference in electronegativity, so their dipole moment is zero. For heteronuclear molecules such as CO, there is a small dipole moment. For HF, there is a larger dipole moment because there is a larger difference in electronegativity.
When a molecule contains more than one bond, the geometry must be taken into account. If the bonds in a molecule are arranged such that their bond moments cancel (vector sum equals zero), then the molecule is nonpolar. This is the situation in CO 2 ( Figure 7.27 ). Each of the bonds is polar, but the molecule as a whole is nonpolar. From the Lewis structure, and using VSEPR theory, we determine that the CO 2 molecule is linear with polar C=O bonds on opposite sides of the carbon atom. The bond moments cancel because they are pointed in opposite directions. In the case of the water molecule ( Figure 7.27 ), the Lewis structure again shows that there are two bonds to a central atom, and the electronegativity difference again shows that each of these bonds has a nonzero bond moment. In this case, however, the molecular structure is bent because of the lone pairs on O, and the two bond moments do not cancel. Therefore, water does have a net dipole moment and is a polar molecule (dipole).
The OCS molecule has a structure similar to CO 2 , but a sulfur atom has replaced one of the oxygen atoms. To determine if this molecule is polar, we draw the molecular structure. VSEPR theory predicts a linear molecule:
The C-O bond is considerably polar. Although C and S have very similar electronegativity values, S is slightly more electronegative than C, and so the C-S bond is just slightly polar. Because oxygen is more electronegative than sulfur, the oxygen end of the molecule is the negative end.
Chloromethane, CH 3 Cl, is a tetrahedral molecule with three slightly polar C-H bonds and a more polar C-Cl bond. The relative electronegativities of the bonded atoms is H < C < Cl, and so the bond moments all point toward the Cl end of the molecule and sum to yield a considerable dipole moment (the molecules are relatively polar).
For molecules of high symmetry such as BF 3 (trigonal planar), CH 4 (tetrahedral), PF 5 (trigonal bipymidal), and SF 6 (octahedral), all the bonds are of identical polarity (same bond moment) and they are oriented in geometries that yield nonpolar molecules (dipole moment is zero). Molecules of less geometric symmetry, however, may be polar even when all bond moments are identical. For these molecules, the directions of the equal bond moments are such that they sum to give a nonzero dipole moment and a polar molecule. Examples of such molecules include hydrogen sulfide, H 2 S (nonlinear), and ammonia, NH 3 (trigonal pyramidal).
To summarize, to be polar, a molecule must:
- Contain at least one polar covalent bond.
- Have a molecular structure such that the sum of the vectors of each bond dipole moment does not cancel.
LINK TO LEARNING
The molecule polarity simulation provides many ways to explore dipole moments of bonds and molecules.
EXAMPLE 7.18
Polarity Simulations
Open the molecule polarity simulation and select the “Three Atoms” tab at the top. This should display a molecule ABC with three electronegativity adjustors. You can display or hide the bond moments, molecular dipoles, and partial charges at the right. Turning on the Electric Field will show whether the molecule moves when exposed to a field, similar to Figure 7.28 .
Use the electronegativity controls to determine how the molecular dipole will look for the starting bent molecule if:
(a) A and C are very electronegative and B is in the middle of the range.
(b) A is very electronegative, and B and C are not.
Solution
(a) Molecular dipole moment points immediately between A and C.
(b) Molecular dipole moment points along the A–B bond, toward A.
Check Your Learning
Determine the partial charges that will give the largest possible bond dipoles.
Answer:
The largest bond moments will occur with the largest partial charges. The two solutions above represent how unevenly the electrons are shared in the bond. The bond moments will be maximized when the electronegativity difference is greatest. The controls for A and C should be set to one extreme, and B should be set to the opposite extreme. Although the magnitude of the bond moment will not change based on whether B is the most electronegative or the least, the direction of the bond moment will.


4.7: Range
- Each type of voice and musical instrument has a characteristic pitch range.
Introduction
The range of a voice or instrument is the set of pitches, from lowest to highest, that it can sing or play. A range can be described using the appropriate octave identification, for example, "from one-line c to two-line g". But it is often easiest to write the range on a staff, as the two notes at the high and low ends of the range.
A piece of music, or one performer's part in that piece, may also be described as having a range, from the lowest to highest note written for the performer in that piece. There is usually a difference (sometimes a large one) between the total range of the part and a smaller range that the part stays in most of the time (heading to the extreme highs and lows only occasionally). This smaller range is called the tessitura of the part. One can also speak of the tessitura of a performer's voice, which is the range in which it sounds the best (so that matching the tessitura of the part and of the performer is a very good idea). Notice the similarity between this second definition and the term power range, sometimes used to describe the most powerful or useful part of an instrument's range.
A register is a distinctive part of a vocal or instrumental range. For example, singers may speak of the head register, in the upper part of their range, and the chest register in the lower part of their range. These two registers sound and feel very different, and the singer may have even have two distinct tessituras, one in each register. The large range of the clarinet is also divided into distinctive registers with different capabilities and very different timbres. Even when an instrument does not have a very large variation in timbre over its range, its players may speak of the difficulty of "playing in the high register" or a "dull timbre in the low register".
Describing a Range
Vocal Ranges
A typical choral arrangement divides women into higher and lower voices and men into higher or lower voices. Most voices can be assigned one of these four ranges, and this gives the composer four vocal lines to work with, which is usually enough. The four main vocal ranges are:
- Soprano – A high female (or boy’s) voice
- Alto – A low female (or boy’s) voice
- Tenor – A high (adult) male voice
- Bass – A low (adult) male voice
Arrangements for these four voices are labelled SATB (for Soprano Alto Tenor Bass). The ranges of the four voices overlap, but singers may find themselves straining or getting an unpleasant sound at the top or a weak sound at the bottom of their ranges. So although the full ranges of an alto and a soprano may look quite similar, the soprano gets a strong, clear sound on the higher notes, and the alto a strong, clear sound in the lower part of the range. But there are vocalists whose strong, best-sounding range falls in a distinctly different place from any of these four voices. The names for some of these ranges are:
- Coloratura Soprano – This is not really a different range from the soprano, but a coloratura soprano has a voice that is unusually high, light, and agile, even for a soprano.
- Mezzo-soprano – In between soprano and alto
- Contralto – Contralto and alto originally referred to the same voice. But some people today use “contralto” to refer to a female voice that is even lower than a typical alto
- Countertenor – A male voice that is unusually high, light, and agile, even for a tenor
- Baritone – A male voice that falls in between tenor and bass
Vocal Ranges
Instrumental Ranges
The same terms used to identify vocal ranges are also often used to identify particular instruments. For example a bass trombone has a lower range than a tenor trombone, and an alto saxophone sounds higher than a tenor saxophone. Some other terms that are used to describe instrument ranges are:
- Contra – Means lower: for example a contrabassoon sounds lower than a regular bassoon, and a contrabass clarinet is even lower than a bass clarinet.
- Piccolo - Means higher (literally "smaller"): for example, a piccolo trumpet is higher than a regular trumpet.
- A Note Name – If an instrument comes in several different sizes (and ranges), the name of a particular size of the instrument may be a note name: for example, an F horn, a B flat clarinet, and a C trumpet. The note name is the name of the fundamental harmonic of the instrument. An instrument with a slightly higher fundamental will have a slightly higher range; an instrument with a much lower fundamental will have a much lower range. Some instruments that are identified this way are transposing instruments, but others are not.
The ranges of some instruments are definite and absolute. For example, even a beginning piano player can play the highest and lowest keys; and even the best player cannot play beyond these. But the ranges of many instruments are, like vocal ranges, not so definite. For example, an experienced horn or clarinet player can play much higher and lower notes than a beginner. An exceptional trumpet player may be able to play - with good sound and technique – high notes that the average high school trumpet player cannot play at all.
Other instruments may be a mix of absolute and indefinite ranges. For example, on any string instrument, nobody can play lower than the note that the lowest string is tuned to. But experienced players can easily play very high notes that inexperienced players have trouble playing well.
So it is sometimes useful to distinguish between a possible range, which includes the notes that a very experienced player can get, and a practical range, that includes all the notes that any competent player (including a good younger player) can get.
Outside of the instrument’s practical range, it may be a strain for even a very good player to play long or tricky passages. So if you are composing or arranging, it’s a very good idea to be able to distinguish between these two ranges for the voices or instruments you include.
Some sources even list the power range of an instrument or voice. This is the part of the range where the instrument or voice is particularly strong. It may be in the middle of the range, or at the top or bottom, but writing in the power range should guarantee that the part is easy to play (or sing), sounds clear and strong, and can be easily heard, even when many other instruments are playing.


4.6: Breach of Contract Example
Breach of Contract
Read this article to learn more about an actual breach of contract.
A funeral home buried a stranger instead of their dad. Now, they're suing.
Byline: Jonathan Edwards
Stacy Holzman wanted to inspect her father's body an hour before his March 2 funeral on Long Island, just to confirm he was wearing the clothes he'd specified -- his favorite Led Zeppelin T-shirt, black jeans and no shoes.
Holzman had a funeral employee drive the hearse into the parking lot, pull out the casket and open it. Inside, she saw the Led Zeppelin shirt and the black jeans. She looked and saw bare feet, just like her father wanted.
But the man wearing the clothes wasn't her father but a stranger. The funeral home employee, however, allegedly insisted that the body belonged to Holzman's father.
On Wednesday, Holzman and her sister, Megan Zaner, sued the Star of David Memorial Chapels and Fletcher Funeral & Cremation Service, accusing the companies of gross negligence and breach of contract. In the 19-page suit filed in the New York Supreme Court of Suffolk County, Holzman and Zaner allege that Fletcher Funeral & Cremation Service transported the wrong body from South Carolina, where their father died, to Long Island, where Star of David Memorial Chapels was supposed to bury him in a Zaner family plot.
Fletcher Funeral & Cremation Service did not respond Thursday to a request for comment. In a statement to The Washington Post, Star of David said Zaner's family had confirmed his identity in New York, allowing the burial to proceed.
Holzman denied that. She said she was stunned by the strange-looking man in her father's clothes and assumed the funeral home employees knew what they were doing, especially after the one who opened the casket allegedly explained that some of the changes in appearance were normal and proceeded with the burial at Mount Ararat Cemetery in Lindenhurst, N.Y.
In the statement, Star of David described what happened as "the mistake made by the funeral home in South Carolina." Still, Star of David said that it's reviewing its procedures and "will make any recommended changes to ensure the correct identification of family members."
"We are committed to continuing to provide the highest level of compassion and care to families who have entrusted us with their loved ones," according to the statement.
Clifford Zaner died on Feb. 25 of respiratory and congestive heart failure, Holzman said. His death was not a shock. He'd had multiple heart attacks and lived with Holzman and her son for a year. At the end of January or early February, he got pneumonia and "his heart just couldn't keep up." He was 72.
Holzman hired Star of David to prepare her father's body and bury it in accordance with Jewish custom, she said. The company then contracted with Fletcher, a funeral home with a location in Fountain Inn, S.C., to transport the body to Long Island, the lawsuit states.
On March 1, Holzman and her 12-year-old son flew from Greenville, S.C., to New York. The next day, they drove to the cemetery, where Holzman said she signed some paperwork and paid Star of David some $12,000 for its services. Then, she asked to see her father's body.
"As the oldest child, it was my responsibility just to peek in and make sure that everything was in order," she told The Post.
When the funeral home employee opened the casket, Holzman saw a man who was clean shaven and had stitches across his forehead, indicating that an autopsy had been performed. Holzman was confused. Her father had always had a mustache and she'd been clear that, in keeping with Jewish custom, her father's body wasn't to be touched.
The employee told her that Star of David routinely shaved facial hair of the bodies that came to them and that an autopsy was also standard when someone died at a hospital, Holzman said.
The employee went inside, saying she had to make some phone calls, Holzman said. About 15 minutes later, the employee re-emerged and allegedly said something like "let's go" before driving to the gravesite, which Holzman took as confirmation that everything was copacetic.
Reluctantly, Holzman was convinced that the man was their father, and they buried him.
About a week-and-a-half later, an employee from Star of David called Holzman while she was out at dinner, she told The Post. He allegedly told Holzman that Fletcher had called to tell them that the man they'd buried wasn't Zaner. His body was still in South Carolina, according to the lawsuit. It had never moved. The next day, Fletcher called Holzman directly to deliver the news "and was very apologetic and said that they made a mistake," she said.
Holzman went to Fletcher to see her father's body. This time, it looked like him -- the facial hair, no mark across his forehead, which was "a bit comforting."
But the gravity of the situation soon occurred to Holzman. A stranger was in the Zaner family plot, in the grave reserved for her father so that he could be buried next to his parents and grandparents, she said. Holzman learned it was not an easy fix, that exhuming the body of the stranger would be "a big deal" and require a court order.
Holzman's lawsuit doesn't provide any information about the man who was buried in her father's plot or what might happen to his body.
After consulting with relatives and myriad rabbis from across the country, they decided to start a new family plot in Jacksonville, Fla., where some of Holzman's relatives live, she said. She arranged to have his body moved on March 22 and two days later "had to endure a second funeral with the proper corpse of their father," her lawsuit states.
Three rabbis were there to oversee the burial, she said. Her son, who'd become close with his grandfather over the year they'd lived together, once again picked up a shovel and, as is Jewish custom, helped fill the grave "until every trace of dirt that was there was taken care of," Holzman told The Post.
"It was gut-wrenching," she added.
Holzman still has family members at Mount Ararat Cemetery. Her great-grandmother died about a month before her dad and was buried in the Zaner family plot. In the coming months, Holzman is supposed to go to her gravesite for an unveiling, a custom in Judaism in which family gathers around a loved one's grave six months to a year after their death to unveil their headstone or, in this case, her footstone.
Holzman said she's dreading it.
"We don't know how we're going to go to New York and go to the cemetery and stand there -- the grave where my dad's supposed to be and isn't."
Copyright: COPYRIGHT 2023 The Washington Post
Source Citation
"A Funeral Home Buried a Stranger Instead of Their Dad. Now, They're Suing." Washingtonpost.com , 21 July 2023, p. NA. Gale In Context: Opposing Viewpoints , link.gale.com/apps/doc/A758014968/OVIC?u=la74598&sid=bookmark-OVIC&xid=4a39e0ae . Accessed 18 Nov. 2024.


3.1: Sets
At its essence, all of mathematics is built on set theory. In this chapter, we will introduce some of the basics of sets and their properties.
Definition 3.1. A set is a collection of objects called elements . If \(A\) is a set and \(x\) is an element of \(A\) , we write \({x\in A}\) . Otherwise, we write \({x\notin A}\) . The set containing no elements is called the empty set , and is denoted by the symbol \({\emptyset}\) . Any set that contains at least one element is referred to as a nonempty set .
If we think of a set as a box potentially containing some stuff, then the empty set is a box with nothing in it. One assumption we will make is that for any set \(A\) , \(A\notin A\) . The language associated to sets is specific. We will often define sets using the following notation, called set-builder notation : \[{S=\{x \in A\mid P(x)\}},\] where \(P(x)\) is some predicate statement involving \(x\) . The first part “ \(x \in A\) " denotes what type of \(x\) is being considered. The predicate to the right of the vertical bar (not to be confused with “divides") determines the condition(s) that each \(x\) must satisfy in order to be a member of the set. This notation is read as “The set of all \(x\) in \(A\) such that \(P(x)\) ." As an example, the set \(\{x\in \mathbb{N}\mid x \mbox{ is even and }x\geq 8\}\) describes the collection of even natural numbers that are greater than or equal to 8.
There are a few sets that are commonly discussed in mathematics and have predefined symbols to denote them. We have already encountered the integers, natural numbers, and real numbers. Notice that our definition of the rational numbers uses set-builder notation.
- Natural Numbers: \({\mathbb{N}:= \{1,2,3,\ldots\}}\) . Some books will include zero in the set of natural numbers, but we do not.
- Integers: \({\mathbb{Z} := \{0, \pm 1, \pm2, \pm 3, \ldots\}}\) .
- Rational Numbers: \({\mathbb{Q} := \{a/b \mid a, b \in \mathbb{Z} \text{ and } b \neq 0\}}\) .
- Real Numbers: \({\mathbb{R}}\) denotes the set of real numbers. We are taking for granted that you have some familiarity with this set.
Since the set of natural numbers consists of the positive integers, the natural numbers are sometimes denoted by \({\mathbb{Z}^+}\) .
Problem 3.2. Unpack the meaning of each of the following sets and provide a description of the elements that each set contains.
- \(A=\{x \in \mathbb{N} \mid x = 3k \mbox{ for some } k\in \mathbb{N} \}\)
- \(B=\{t \in \mathbb{R} \mid t \leq 2 \mbox{ or } t\geq 7 \}\)
- \(C=\{t \in \mathbb{Z} \mid t^2 \leq 2 \}\)
- \(D=\{s \in \mathbb{Z} \mid -3<s\leq 5 \}\)
- \(E=\{m \in \mathbb{R} \mid m = 1 - \frac{1}{n} \mbox{, where } n \in \mathbb{N} \}\)
Problem 3.3. Write each of the following sentences using set-builder notation.
- [prob:set-builder examples a] The set of all real numbers less than \(-\sqrt{2}\) .
- [prob:set-builder examples b] The set of all real numbers greater than \(-12\) and less than or equal to 42.
- The set of all even integers.
Parts (a) and (b) of Problem 3.3 are examples of intervals.
Definition 3.4. For \(a,b\in\mathbb{R}\) with \(a<b\) , we define the following sets, referred to as intervals .
- (a,b) := {x ∈ R | a<x<b}
- [a,b] := {x ∈ R | a\(\leq\) x \(\leq\) b}
- [a,b) := {x ∈ R | a \(\leq\) x< b}
- (a,\(\infty\)) := {x ∈ R | a<x}
- (- \(\infty\) ,b) := {x ∈ R | x<b}
- (- \(\infty\) , \(\infty\) ) := R
We analogously define \({(a,b]}\) , \({[a,\infty)}\) , and \({(-\infty,b]}\) . Intervals of the form \((a,b)\) , \((-\infty,b)\) , \((a,\infty)\) , and \((-\infty,\infty)\) are called open intervals while \([a,b]\) is referred to as a closed interval . A bounded interval is any interval of the form \((a,b)\) , \([a,b)\) , \((a,b]\) , and \([a,b]\) . For bounded intervals, \(a\) and \(b\) are called the endpoints of the interval.
We will always assume that any time we write \((a,b), [a,b], (a,b]\) , or \([a,b)\) that \(a<b\) . We will see where the terminology of “open" and “closed" comes from in Section 5.2.
Problem 3.5. Give an example of each of the following.
- An interval that is neither an open nor closed interval.
- An infinite set that is not an interval.
Definition 3.5. If \(A\) and \(B\) are sets, then we say that \(A\) is a subset of \(B\) , written \({A\subseteq B}\) , provided that every element of \(A\) is an element of \(B\) .
Problem 3.7. List all of the subsets of \(A=\{1,2,3\}\) .
Every nonempty set always has two subsets. Notice that if \(A=\emptyset\) , then Parts (a) and (b) of the next theorem say the same thing.
Theorem 3.8. Let \(A\) be a set. Then
- \(A\subseteq A\) , and
- \(\emptyset \subseteq A\) .
Observe that “ \(A\subseteq B\) " is equivalent to “For all \(x\) (in the universe of discourse), if \(x\in A\) , then \(x\in B\) ." Since we know how to deal with “for all" statements and conditional propositions, we know how to go about proving \(A\subseteq B\) . If \(A\) happens to be the empty set, then the statement “For all \(x\) (in the universe of discourse), if \(x\in A\) , then \(x\in B\) " is vacuously true. This is in agreement with Theorem 3.8(b), which states that the empty set is always a subset of every set. In light of this, it is common to omit discussion of the case when \(A\) is the empty set when proving that \(A\) is s a subset of \(B\) .
Problem 3.9. Suppose \(A\) and \(B\) are sets. Describe a skeleton proof for proving that \(A\subseteq B\) .
Theorem 3.10. Suppose that \(A\) , \(B\) , and \(C\) are sets. If \(A\subseteq B\) and \(B\subseteq C\) , then \(A\subseteq C\) .
Definition 3.11. Two sets \(A\) and \(B\) are equal , denoted \({A=B}\) , if the sets contain the same elements.
Since the next theorem is a biconditional proposition, you need to write two distinct subproofs, one for “ \(A=B\) implies \(A \subseteq B\) and \(B \subseteq A\) ", and another for “ \(A \subseteq B\) and \(B \subseteq A\) implies \(A=B\) ". Be sure to make it clear to the reader when you are proving each implication.
Theorem 3.12. Suppose that \(A\) and \(B\) are sets. Then \(A=B\) if and only if \(A \subseteq B\) and \(B \subseteq A\) .
Note that if we want to prove \(A=B\) , then we have to do two separate subproofs: one for \(A\subseteq B\) and one for \(B\subseteq A\) . Be sure to make it clear to the reader where these subproofs begin and end. One approach is to label each subproof with “ \((\subseteq)\) " and “ \((\supseteq)\) " (including the parentheses), respectively.
Definition 3.13. If \(A\subseteq B\) , then \(A\) is called a proper subset provided that \(A\neq B\) . In this case, we may write \({A\subset B}\) or \({A\subsetneq B}\) .
Note that some authors use \(\subset\) to mean \(\subseteq\) , so some confusion could arise if you are not reading carefully.
Definition 3.14. Let \(A\) and \(B\) be sets in some universe of discourse \(U\) .
- The union of the sets \(A\) and \(B\) is \(A \cup B\) := { \(x\in U \mid x\in A \mbox{ or } x\in B \)} .
- The intersection of the sets \(A\) and \(B\) is \(A \cap B \) := { \(x\in U \mid x\in A \mbox{ and } x\in B\) } .
- The set difference of the sets \(A\) and \(B\) is \(A \setminus B\) := { \(x\in U \mid x\in A \mbox{ and } x\notin B \)} .
- The complement of \(A\) (relative to \(U\) ) is the set \(A^c\):= \( U \setminus A\) ={\( x \in U \mid x \notin A\)} .
Definition 3.15. If two sets \(A\) and \(B\) have the property that \(A \cap B = \emptyset\) , then we say that \(A\) and \(B\) are disjoint sets.
Problem 3.16. Suppose that the universe of discourse is \(U=\{1,2,3,4,5,6,7,8,9,10\}\) . Let \(A=\{1, 2, 3, 4, 5\}\) , \(B=\{1, 3, 5\}\) , and \(C=\{2, 4, 6, 8\}\) . Find each of the following.
- \(A \cap C\)
- \(B \cap C\)
- \(A \cup B\)
- \(A\setminus B\)
- \(B \setminus A\)
- \(C \setminus B\)
- \(B^c\)
- \(A^c\)
- \((A\cup B)^c\)
- \(A^c\cap B^c\)
Problem 3.17. Suppose that the universe of discourse is \(U=\mathbb{R}\) . Let \(A=[-3,-1)\) , \(B=(-2.5,2)\) , and \(C=(-2,0]\) . Find each of the following.
- \(A^c\)
- \(A \cap C\)
- \(A \cap B\)
- \(A \cup B\)
- \((A\cap B)^c\)
- \((A\cup B)^c\)
- \(A \setminus B\)
- \(A\setminus (B \cup C)\)
- \(B \setminus A\)
Problem 3.18. Suppose that the universe of discourse is \(U=\{x,y, z, \{y\}, \{x,z\}\}\) . Let \(S=\{x,y,z\}\) and \(T=\{x,\{y\}\}\) . Find each of the following.
- \(S \cap T\)
- \((S\cup T)^{c}\)
- \(T\setminus S\)
Theorem 3.19. If \(A\) and \(B\) are sets such that \(A \subseteq B\) , then \(B^c \subseteq A^c\) .
Theorem 3.20. If \(A\) and \(B\) are sets, then \(A\setminus B = A \cap B^c\) .
In Chapter 2, we encountered De Morgan’s Law (see Theorem 2.26 and Problem 2.27), which provided a method for negating compound propositions involving conjunctions and disjunctions. The next theorem provides a method for taking the complement of unions and intersections of sets. This result is also known as De Morgan’s Law. Do you see why?
Theorem 3.21. If \(A\) and \(B\) are sets, then
- \((A \cup B)^c = A^c \cap B^c\) , and
- \((A \cap B)^c = A^c \cup B^c\) .
The next theorem indicates how intersections and unions interact with each other.
Theorem 3.22. If \(A\) , \(B\) , and \(C\) are sets, then
- \(A \cup(B\cap C) = (A\cup B)\cap (A\cup C)\) , and
- \(A\cap (B\cup C)= (A\cap B)\cup (A\cap C)\) .
Problem 3.23. For each of the statements (a)–(d) on the left, find an equivalent symbolic proposition chosen from the list (i)–(v) on the right. Note that not every statement on the right will get used.
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4.5: Treatment
An “alternative to incarceration” is any kind of punishment other than time in prison or jail that can be given to a person who commits a crime. Frequently, punishments other than prison or jail time place serious demands on offenders and provide them with intensive court and community supervision. Just because a certain punishment does not involve time in prison or jail does not mean it is “soft on crime” or a “slap on the wrist.” Alternatives to incarceration can repair harms suffered by victims, provide benefits to the community, treat the drug-addicted or mentally ill, and rehabilitate offenders. Alternatives can also reduce prison and jail costs and prevent additional crimes in the future. Before we can maximize the benefits of alternatives to incarceration, however, we must repeal mandatory minimums and give courts the power to use cost-effective, recidivism-reducing sentencing options instead.


17.1C: Primary and Secondary Metabolites
- Describe how primary and secondary metabolites can be used in industrial microbiology to obtain amino acids, develop vaccines and antibiotics, and isolate chemicals for organic synthesis
Bacterial metabolism can be classified into three major categories: the kind of energy used for growth, the carbon source, and the electron donors used for growth. Pathogenic bacteria are capable of exhibiting various types of metabolism. Metabolites, the intermediates and products of metabolism, are typically characterized by small molecules with various functions. Metabolites can be categorized into both primary and secondary metabolites. These metabolites can be used in industrial microbiology to obtain amino acids, develop vaccines and antibiotics, and isolate chemicals necessary for organic synthesis.
Primary Metabolites
Primary metabolites are involved in growth, development, and reproduction of the organism. The primary metabolite is typically a key component in maintaining normal physiological processes; thus, it is often referred to as a central metabolite. Primary metabolites are typically formed during the growth phase as a result of energy metabolism, and are deemed essential for proper growth. Examples of primary metabolites include alcohols such as ethanol, lactic acid, and certain amino acids. Within the field of industrial microbiology, alcohol is one of the most common primary metabolites used for large-scale production. Specifically, alcohol is used for processes involving fermentation which produce products like beer and wine. Additionally, primary metabolites such as amino acids– including L-glutamate and L-lysine, which are commonly used as supplements– are isolated via the mass production of a specific bacterial species, Corynebacteria glutamicum . Another example of a primary metabolite commonly used in industrial microbiology includes citric acid. Citric acid, produced by Aspergillus niger , is one of the most widely used ingredients in food production. It is commonly used in pharmaceutical and cosmetic industries as well.
Secondary Metabolites
Secondary metabolites are typically organic compounds produced through the modification of primary metabolite synthases. Secondary metabolites do not play a role in growth, development, and reproduction like primary metabolites do, and are typically formed during the end or near the stationary phase of growth. Many of the identified secondary metabolites have a role in ecological function, including defense mechanism(s), by serving as antibiotics and by producing pigments. Examples of secondary metabolites with importance in industrial microbiology include atropine and antibiotics such as erythromycin and bacitracin. Atropine, derived from various plants, is a secondary metabolite with important use in the clinic. Atropine is a competitive antagonist for acetycholine receptors, specifically those of the muscarinic type, which can be used in the treatment of bradycardia. Antibiotics such as erythromcyin and bacitracin are also considered to be secondary metabolites. Erythromycin, derived from Saccharopolyspora erythraea , is a commonly used antibiotic with a wide antimicrobial spectrum. It is mass produced and commonly administered orally. Lastly, another example of an antibiotic which is classified as a secondary metabolite is bacitracin. Bacitracin, derived from organisms classified under Bacillus subtilis , is an antibiotic commonly used a topical drug. Bacitracin is synthesized in nature as a nonribosomal peptide synthetase that can synthesize peptides; however, it is used in the clinic as an antibiotic.
Key Points
- Primary metabolites are considered essential to microorganisms for proper growth.
- Secondary metabolites do not play a role in growth, development, and reproduction, and are formed during the end or near the stationary phase of growth.
- These metabolites can be used in industrial microbiology to obtain amino acids, develop vaccines and antibiotics, and isolate chemicals necessary for organic synthesis.
Key Terms
- bradycardia : the slowing of the heartbeat to below average


10: Linear Momentum and the Center of Mass
- Understand how to calculate linear momentum.
- Understand how to calculate impulse and that it corresponds to a change in momentum.
- Understand when and how to apply conservation of linear momentum to model situations.
- Understand the difference between elastic and inelastic collisions, and when mechanical energy is conserved.
- Understand how to calculate the center of mass of an object.
In this chapter, we introduce the concepts of linear momentum and of center of mass. Momentum is a quantity that, like energy, can be defined from Newton’s Second Law, to facilitate building models. Since momentum is often a conserved quantity within a system, it can make calculations much easier than using forces. The concepts of momentum and of center of mass will also allow us to apply Newton’s Second Law to systems comprised of multiple particles including solid objects.
You hit a pool ball square on with the cue ball. If both balls have the same mass, and you can neglect any “english” on the cue ball, what happens to the cue ball?
- It stops.
- It continues, with half of its original speed.
- It continues, with its original speed.
- It rebounds, with its original speed.


9.4: Additional Campus Resources
Advising
Every student at HACC is assigned an academic advisor. The College has both professional advisors and faculty advisors. Professional advisors have a large caseload of students, and their primary role consists of advising. Faculty advisors have a smaller caseload of students since their primary focus is teaching. Meeting with your advisor every semester is recommended to plan your classes for the following semester and into the future. These meetings will ensure you stay on track and that you are taking the appropriate classes. You can find your advisor’s name and contact information by logging into myHACC and looking under the light blue-gray box under “Academic Standing Information” on the right side of the screen or by clicking on "Student Profile" and look for your "Primary/Program Advisor" near the center of the page. Because advising appointments fill up as registration deadlines approach, you are encouraged to reach out to your advisor well in advance of registration. For example, if registration begins at the beginning of October, reach out in early September to schedule a meeting with your advisor.
Link to Advising: HACC Advising [www.hacc.edu]
HACC’s Consultation, Advocacy, Referrals and Empowerment (CARE) Center
HACC cares about the success and well-being of students, and the CARE Center is here to help. HACC’s first priority is providing a high-quality, affordable, accessible education to all. However, some students need support outside the classroom in order to succeed inside the classroom.
HACC proudly established the Consultation, Advocacy, Referrals and Empowerment (CARE) Center as a single stop where students can go for assistance with non-academic challenges. This is a free resource available to all current HACC students. HACC is committed to providing excellent services to students no matter where they are.
Dedicated staff in HACC’s CARE Center will connect students to resources at the College and in the community to help with the following:
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Child care |
Students may be referred to the KEYS Program for access to the Early Learning Resource Center and financially subsidized child care, or students are referred to community-based resources according to their region. Child care options are available at HACC’s Harrisburg and Lancaster campuses. |
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Food |
The CARE Center maintains a database of food pantries, food banks and food distribution locations in HACC’s service area. Coordinators provide information and referrals to locations for students in need. |
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Health care |
The CARE Center has established relationships with hospitals and clinics. Coordinators refer students to services and make them aware of low-cost and pro-bono healthcare services as much as possible. |
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Housing |
The CARE Center maintains a database of shelters and housing assistance agencies in HACC’s service area. Coordinators refer students in need of housing to agencies and shelters. |
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Mental health services |
If a coordinator meets with a student who is in crisis and requires mental health treatment, the coordinator refers the student to Mazzitti & Sullivan EAP Services . |
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Social services |
Coordinators refer students in need of social services – including Temporary Assistance for Needy Families (TANF) and Supplemental Nutrition Assistance Program (SNAP) – to the KEYS Program . |
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Transportation |
CARE Center staff coordinate with the College’s bookstore to assist students with public transportation bus passes. |
Link to CARE Center Information: HACC CARE Center [www.hacc.edu]
Career Services at HACC
Career Services is here to help you from the start of your college journey and beyond. Career Services can assist you with many tasks including:
- Resume Writing
- Cover Letter Writing
- Mock Interviewing
- Job Shadowing & Internship Opportunities
- Major Exploration (for those undecided)
- Job Opportunities through College Central Network
College Central Network (CCN) is available to students, alumni and employers. Full-time, part-time, seasonal, on-campus employment, internships and volunteer positions are posted. Valuable resources like podcasts, a resume builder, and Mentor Network are great tools you can use on CCN. College Central [www.collegecentral.com]
For more information and resources, please visit the Career Services website at: HACC Career Services [www.hacc.edu]
The HACC College Calendar
Knowing where to access the College’s calendar and having an understanding of semester start and end dates, break periods and deadlines is important to student success. To access the College calendar, please go to hacc.edu and click on the “Current Students” tab. Then select “College Calendars.” Make sure you select the appropriate year since several calendars will be displayed.
Link to access the College’s calendars: HACC College Calendars [www.hacc.edu]
Emergency Assistance Fund for HACC Students
What are emergency assistance funds?
Students can apply for emergency assistance funds if they are facing extraordinary financial circumstances. Learn more about these funds.
What can students use the funds for?
- Books
- Testing fees (for example, TEAS exams)
- Course-required items from HACC bookstores (for example, uniforms, equipment and supplies)
- Physicals and medical tests (for example, tuberculosis tests)
- Emergency auto repair expenses to attend in-person activities related to instruction (for example, clinicals and classes held on a HACC campus)
- Bus passes
- Childcare
- Personal home catastrophe (for example, fire or flood that directly affects your ability to attend classes)
Emergency assistance funds do not cover all expenses. Please see the link below for more information on who is eligible and how you can apply.
Link to Emergency Assistance Fund Information: HACC Paying for School [www.hacc.edu]
e2Campus Alerts
With e2Campus, you’ll be informed quickly when the College has immediate announcements, such as weather-related closings, delays, cancelations or emergencies. Notifications of these events will come right to your cell phone or email. To sign up for these alerts, please use the link below. HACC Public Safety and Security [www.hacc.edu]
Financial Aid
HACC’s Financial Aid Office is here to assist you and your family by providing information on ways to pay for your education. The Financial Aid Office can:
- Help you complete the Free Application for Federal Student Aid (FAFSA)
- Help you complete any financial aid paperwork you may need
- Provide advice on borrowing student loans
- Answer any question you have about the financial aid process
Make sure to check the Financial Aid tab on myHACC regularly and your HAWKMail to receive messages from the Financial Aid Office and to see any paperwork that might be required.
HAWKMail (HACC Student Email)
Your HACC email account also is referred to as HAWKMail, and checking HAWKMail regularly is key. Your HACC email account is what your instructors and the College will use to contact you and communicate with you. Some students have their HACC email forwarded to their personal email. Information on how to do this can be found in the link below.
HACC Email Information Link: HACC HAWKMail [www.hacc.edu]
Instructor Office Hours
On your course syllabus, you should see office hours listed for your instructor. Office hours are open times that professors hold for meeting with students and do not require appointments. Every instructor treats office hours in a different manner. Some instructors may hold virtual office hours, while others may have in-person hours. Please check your syllabus for specific times and locations. When you struggle in a class or have questions, seeking help from the instructor is an important action to take.
Learning Commons at HACC
Each HACC campus has a Learning Commons where students are able to access student success services including the library, tutoring, testing, and technology support in one location.
Tutoring Resources at HACC
HACC offers free tutoring resources for all HACC students. In-person, remote and online tutoring options are available. Any student at HACC may access in-person, remote or online tutoring regardless of whether you are taking classes at a physical HACC location, remote in Zoom or online. Tutoring services also include an online writing lab, study groups, and academic coaching.
Use the link below to find out more about online and in-person tutoring options: HACC Library Resources [libguides.hacc.edu]
Library Resources at HACC
The library has access to numerous resources including articles and newspapers in online databases, ebooks, streaming videos and media, as well as print books, and journals. The faculty librarians can help you navigate these resources and select materials that best meet your research needs. In several of your classes, a librarian may be involved to teach about information literacy. You can schedule a research appointment with a librarian via the library website below. The library also has access to online chat 24/7. Within the campus space, there are areas available for quiet as well as collaborative work. Computers are equipped with technologies you might use in a class, and printing is available. You also are able to check out equipment and devices such as laptops, iPads, and calculators.
Link to Library Resources: HACC Library Resources [libguides.hacc.edu]
Testing Resources at HACC
The Testing Center in the Learning Commons provides a secure environment where students can take placement exams, specialized tests such as NCLEX or CLEP, as well as academic make-up exams when approved by your instructor. The Testing Center has both in person and online options available.
Technology User Support
User Support offers technology assistance to students.You can drop in for a Zoom session, and the IT call center is available 24/7. If you do not have access to a computer, HACC has a technology and internet access loan program that allows students to borrow equipment for their classes. You can find out more about this program on the User Support web link below.
Link to Technology User Support: HACC Library [www.hacc.edu]
Mental Health Resources
HACC will pay for up to three counseling sessions per semester for a student in need of services. Our partner agency, Mazzitti & Sullivan EAP Services, Inc., provides comprehensive clinical mental health services to HACC students who are currently enrolled in credit classes.
This agreement allows students to access:
- A robust, national network of licensed and insured therapists
- Confidential help for a variety of mental health concerns, including depression, anxiety, stress, and relationship issues
- Options for face-to-face, telephonic, and on-demand virtual counseling
- Triage services 24 hours a day, seven days a week
Phone Options
- Call 1-800-543-5080 to speak to someone 24/7 or to schedule an appointment.
- Call 1-855-264-3248 if you use TeleTYpe (TTY)/Telecommunications Device for the Deaf (TDD.)
Online Option
- Go to Mazzitti and Sullivan Get Started [mseap.com] .
- Look for the "Employee" box and click "Complete this Form."
- When completing the form, under "Organization or Group Sponsoring Your Program," indicate you are a student.
- Click "Submit" at the bottom of the form.
- Expect your message to be returned by the end of the next business day.
Link to Mental Health Resources at HACC: HACC Wellness Resources [www.hacc.edu]
Scholarships for HACC Students
Did you know the HACC Foundation offers over 200 scholarships specifically for our students? Scholarship Information Link: HACC Paying for School [www.hacc.edu]
Some commonly asked questions can be viewed below:
Question |
Answer |
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What are HACC Foundation scholarships? |
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What kinds of scholarships does the HACC Foundation offer students? |
HACC offers scholarships for both credit and noncredit program students. Over 200 individualized opportunities are available for both current and future students. |
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Information on (non-HACC) private scholarships |
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Applying for credit or noncredit scholarships |
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Student Access (formerly Disability) Services
Student Access Services is committed to an equal educational opportunity and full inclusion for students requiring accommodations for disabilities, temporary or permanent.
What is the purpose of Student Access Services?
Student Access Services has multiple purposes.
- Help students achieve educational objectives
- Offer support services
- Provide academic accommodations
- Promote a supportive learning environment
- Promote student independence, program accessibility, and a psychologically-supportive environment
How can students receive Student Access Services?
Contact Student Access Services to submit documentation of your disability. Once the documentation is received, an intake meeting will be scheduled with you. During this meeting, appropriate accommodations will be determined. Your assessment report must:
- Be completed by a qualified professional. These include medical doctors or licensed psychologists
- Be completed within the last three years
- Contain the evaluator's name, title, and assessment date
- Identify the disability, describe the limits it imposes and include recommended accommodations
Link to Student Access Services Information: HACC Access Services [www.hacc.edu]
Student Involvement
Get involved in campus life by joining student government or a club or organization. Students who participate in student organizations usually are more satisfied with their college experience. By getting involved, you have the opportunity to:
- learn more about HACC
- meet a variety of people
- develop leadership skills
For more information about student government, clubs and activities at HACC, please visit the Student Involvement website below.
Link to Student Involvement: HACC Student Involvement [www.hacc.edu]
Transferring Services
According to four-year colleges and universities, students who complete their associate degrees at HACC are more successful than students who begin at the university as first-year students. Moreover, depending on the four-year institution, you can save thousands of dollars by completing your first two years at HACC. See this link for why you should complete your degree at HACC: HACC Programs and Courses [www.hacc.edu]
Transfer Resources Link: HACC Transfer Services [www.hacc.edu]
Using HACC Resources
Assume you are waiting for a class to start and chatting with a classmate. The classmate seems a little distracted and perhaps even upset today, and you ask if anything is wrong. Your classmate says he might be losing his job since the store where he works is closing. He’s worried that he might not be able to complete a paper that’s due in this class next week since he’s going to have to start searching for another job. Think about the HACC resources explained in this chapter.
- What resource could help your classmate find another job?
- What resource could help him seek additional funding for school?
- What resource might be able to assist with paying for groceries?
- What resource would you recommend for him to communicate concerns about this situation to his instructors?