41.4B: Muscles
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Animals use muscles to convert the chemical energy of ATP into mechanical work. Three different kinds of muscles are found in vertebrate animals.
- Heart muscle also called cardiac muscle makes up the wall of the heart. Throughout our life, it contracts some 70 times per minute pumping about 5 liters of blood each minute.
-
Smooth muscle
is found in the walls of all the hollow organs of the body (except the heart). Its contraction reduces the size of these structures. Thus it
- regulates the flow of blood in the arteries
- moves your breakfast along through your gastrointestinal tract
- expels urine from your urinary bladder
- sends babies out into the world from the uterus
- regulates the flow of air through the lungs
- Skeletal muscle , as its name implies, is the muscle attached to the skeleton. It is also called striated muscle . The contraction of skeletal muscle is under voluntary control.
Anatomy of Skeletal Muscle
A single skeletal muscle, such as the triceps muscle, is attached at its
- origin to a large area of bone; in this case, the humerus .
- at its other end, the insertion , it tapers into a glistening white tendon which, in this case, is attached to the ulna , one of the bones of the lower arm.
As the triceps contracts, the insertion is pulled toward the origin and the arm is straightened or extended at the elbow. Thus the triceps is an extensor . Because skeletal muscle exerts force only when it contracts, a second muscle - a flexor - is needed to flex or bend the joint. The biceps muscle is the flexor of the lower arm. Together, the biceps and triceps make up an antagonistic pair of muscles. Similar pairs, working antagonistically across other joints, provide for almost all the movement of the skeleton.
The Muscle Fiber
Skeletal muscle is made up of thousands of cylindrical muscle fibers often running all the way from origin to insertion. The fibers are bound together by connective tissue through which run blood vessels and nerves.Each muscle fibers contains:
- an array of myofibrils that are stacked lengthwise and run the entire length of the fiber;
- mitochondria ;
- an extensive smooth endoplasmic reticulum (SER) ;
- many nuclei (thus each skeletal muscle fiber is a syncytium).
The multiple nuclei arise from the fact that each muscle fiber develops from the fusion of many cells (called myoblasts ). The number of fibers is probably fixed early in life. This is regulated by myostatin , a cytokine that is synthesized in muscle cells (and circulates as a hormone later in life). Myostatin suppresses skeletal muscle development. (Cytokines secreted by a cell type that inhibit proliferation of that same type of cell are called chalones .) Cattle and mice with inactivating mutations in their myostatin genes develop much larger muscles. Some athletes and other remarkably strong people have been found to carry one mutant myostatin gene. These discoveries have already led to the growth of an illicit market in drugs supposedly able to suppress myostatin.
In adults, increased muscle mass comes about through an increase in the thickness of the individual fibers and increase in the amount of connective tissue. In the mouse, at least, fibers increase in size by attracting more myoblasts to fuse with them. The fibers attract more myoblasts by releasing the cytokine interleukin 4 ( IL-4 ). Anything that lowers the level of myostatin also leads to an increase in fiber size.
Because a muscle fiber is not a single cell, its parts are often given special names such as
- sarcolemma for plasma membrane
- sarcoplasmic reticulum for endoplasmic reticulum
- sarcosomes for mitochondria
- sarcoplasm for cytoplasm
Although this tends to obscure the essential similarity in structure and function of these structures and those found in other cells.
The nuclei and mitochondria are located just beneath the plasma membrane. The endoplasmic reticulum extends between the myofibrils. Seen from the side under the microscope, skeletal muscle fibers show a pattern of cross banding, which gives rise to the other name: striated muscle . The striated appearance of the muscle fiber is created by a pattern of alternating dark A bands and light I bands .
- The A bands are bisected by the H zone running through the center of which is the M line .
- The I bands are bisected by the Z disk .
Each myofibril is made up of arrays of parallel filaments .
- The thick filaments have a diameter of about 15 nm. They are composed of the protein myosin .
- The thin filaments have a diameter of about 5 nm. They are composed chiefly of the protein actin along with smaller amounts of two other proteins - troponin and tropomyosin .
The anatomy of a sarcomere
The entire array of thick and thin filaments between the Z disks is called a sarcomere .
- The thick filaments produce the dark A band .
- The thin filaments extend in each direction from the Z disk. Where they do not overlap the thick filaments, they create the light I band .
- The H zone is that portion of the A band where the thick and thin filaments do not overlap.
- The M line runs through the exact center of the sarcomere. Molecules of the giant protein, titin , extend from the M line to the Z disk. One of its functions is to provide elasticity to the muscle. It also provides a scaffold for the assembly of a precise number of myosin molecules in the thick filament (294 in one case). It may also dictate the number of actin molecules in the thin filaments.
Shortening of the sarcomeres in a myofibril produces the shortening of the myofibril and, in turn, of the muscle fiber of which it is a part. [This electron micrograph of a single sarcomere was kindly provided by Dr. H. E. Huxley.]
Activation of Skeletal Muscle
The contraction of skeletal muscle is controlled by the nervous system. The Dying Lioness (an Assyrian relief dating from about 650 B.C.) shows this vividly. Injury to the spinal cord has paralyzed the otherwise undamaged hind legs. In this respect, skeletal muscle differs from smooth and cardiac muscle. Both cardiac and smooth muscle can contract without being stimulated by the nervous system. Nerves of the autonomic branch of the nervous system lead to both smooth and cardiac muscle, but their effect is one of moderating the rate and/or strength of contraction.
The Neuromuscular Junction
Nerve impulses (action potentials) traveling down the motor neurons of the sensory-somatic branch of the nervous system cause the skeletal muscle fibers at which they terminate to contract. The junction between the terminal of a motor neuron and a muscle fiber is called the neuromuscular junction. It is simply one kind of synapse. (The neuromuscular junction is also called the myoneural junction.)
The terminals of motor axons contain thousands of vesicles filled with acetylcholine ( ACh ). Many of these can be seen in the electron micrograph on the left (courtesy of Prof. B. Katz).
When an action potential reaches the axon terminal, hundreds of these vesicles discharge their ACh onto a specialized area of postsynaptic membrane on the muscle fiber (the folded membrane running diagonally upward from the lower left). This area contains a cluster of transmembrane channels that are opened by ACh and let sodium ions (Na + ) diffuse in.
The interior of a resting muscle fiber has a resting potential of about −95 mV. The influx of sodium ions reduces the charge, creating an end plate potential . If the end plate potential reaches the threshold voltage (approximately −50 mV), sodium ions flow in with a rush and an action potential is created in the fiber. The action potential sweeps down the length of the fiber just as it does in an axon. No visible change occurs in the muscle fiber during (and immediately following) the action potential. This period, called the latent period , lasts from 3–10 msec.
Before the latent period is over,
-
the enzyme
acetylcholinesterase
- breaks down the ACh in the neuromuscular junction (at a speed of 25,000 molecules per second)
- the sodium channels close, and
- the field is cleared for the arrival of another nerve impulse.
- the resting potential of the fiber is restored by an outflow of potassium ions .
The brief (1–2 msec) period needed to restore the resting potential is called the refractory period .
Tetanus
The process of contracting takes some 50 msec; relaxation of the fiber takes another 50–100 msec. Because the refractory period is so much shorter than the time needed for contraction and relaxation, the fiber can be maintained in the contracted state so long as it is stimulated frequently enough (e.g., 50 stimuli per second). Such sustained contraction is called tetanus .
In the above figure:
- When shocks are given at 1/sec, the muscle responds with a single twitch.
- At 5/sec and 10/sec, the individual twitches begin to fuse together, a phenomenon called clonus .
- At 50 shocks per second, the muscle goes into the smooth, sustained contraction of tetanus.
Clonus and tetanus are possible because the refractory period is much briefer than the time needed to complete a cycle of contraction and relaxation. Note that the amount of contraction is greater in clonus and tetanus than in a single twitch.
As we normally use our muscles, the individual fibers go into tetanus for brief periods rather than simply undergoing single twitches.
The Sliding-Filament Model
Each molecule of myosin in the thick filaments contains a globular subunit called the myosin head . The myosin heads have binding sites for the actin molecules in the thin filaments and ATP. Activation of the muscle fiber causes the myosin heads to bind to actin. An allosteric change occurs which draws the thin filament a short distance (~10 nm) past the thick filament. Then the linkages break (for which ATP is needed) and reform farther along the thin filament to repeat the process. As a result, the filaments are pulled past each other in a ratchetlike action. There is no shortening, thickening, or folding of the individual filaments.
Electron microscopy supports this model.
As a muscle contracts,
- the Z disks come closer together
- the width of the I bands decreases
- the width of the H zones decreases
- there is no change in the width of the A band
Conversely, as a muscle is stretched,
- the width of the I bands and H zones increases
- but there is still no change in the width of the A band
Coupling Excitation to Contraction
Calcium ions (Ca 2+ ) link action potentials in a muscle fiber to contraction.
- In resting muscle fibers, Ca 2+ is stored in the endoplasmic (sarcoplasmic) reticulum.
- Spaced along the plasma membrane ( sarcolemma ) of the muscle fiber are inpocketings of the membrane that form "T-tubules" . These tubules plunge repeatedly into the interior of the fiber.
- The T-tubules terminate near the calcium-filled sacs of the sarcoplasmic reticulum .
- Each action potential created at the neuromuscular junction sweeps quickly along the sarcolemma and is carried into the T-tubules.
- The arrival of the action potential at the ends of the T-tubules triggers the release of Ca 2+ .
- The Ca 2+ diffuses among the thick and thin filaments where it
- binds to troponin on the thin filaments.
- This turns on the interaction between actin and myosin and the sarcomere contracts.
- Because of the speed of the action potential (milliseconds), the action potential arrives virtually simultaneously at the ends of all the T-tubules, ensuring that all sarcomeres contract in unison.
- When the process is over, the calcium is pumped back into the sarcoplasmic reticulum using a Ca 2+ ATPase .
Isotonic versus Isometric Contractions
If a stimulated muscle is held so that it cannot shorten, it simply exerts tension. This is called an isometric ("same length") contraction. If the muscle is allowed to shorten, the contraction is called isotonic ("same tension").
Motor Units
All motor neurons leading to skeletal muscles have branching axons, each of which terminates in a neuromuscular junction with a single muscle fiber. Nerve impulses passing down a single motor neuron will thus trigger contraction in all the muscle fibers at which the branches of that neuron terminate. This minimum unit of contraction is called the motor unit .
The size of the motor unit is small in muscles over which we have precise control. Examples:
- a single motor neuron triggers fewer than 10 fibers in the muscles controlling eye movements
- the motor units of the muscles controlling the larynx are as small as 2–3 fibers per motor neuron
- In contrast, a single motor unit for a muscle like the gastrocnemius (calf) muscle may include 1000–2000 fibers (scattered uniformly through the muscle).
Although the response of a motor unit is all-or-none, the strength of the response of the entire muscle is determined by the number of motor units activated. Even at rest, most of our skeletal muscles are in a state of partial contraction called tonus . Tonus is maintained by the activation of a few motor units at all times even in resting muscle. As one set of motor units relaxes, another set takes over.
Fueling Muscle Contraction
ATP is the immediate source of energy for muscle contraction. Although a muscle fiber contains only enough ATP to power a few twitches, its ATP "pool" is replenished as needed. There are three sources of high-energy phosphate to keep the ATP pool filled.
- creatine phosphate
- glycogen
- cellular respiration in the mitochondria of the fibers.
Creatine phosphate
The phosphate group in creatine phosphate is attached by a "high-energy" bond like that in ATP. Creatine phosphate derives its high-energy phosphate from ATP and can donate it back to ADP to form ATP.
Creatine phosphate + ADP ↔ creatine + ATP
The pool of creatine phosphate in the fiber is about 10 times larger than that of ATP and thus serves as a modest reservoir of ATP.
Glycogen
Skeletal muscle fibers contain about 1% glycogen . The muscle fiber can degrade this glycogen by glycogenolysis producing glucose-1-phosphate. This enters the glycolytic pathway to yield two molecules of ATP for each pair of lactic acid molecules produced. Not much, but enough to keep the muscle functioning if it fails to receive sufficient oxygen to meet its ATP needs by respiration.
However, this source is limited and eventually the muscle must depend on cellular respiration.
Cellular respiration
Cellular respiration not only is required to meet the ATP needs of a muscle engaged in prolonged activity (thus causing more rapid and deeper breathing), but is also required afterwards to enable the body to resynthesize glycogen from the lactic acid produced earlier (deep breathing continues for a time after exercise is stopped). The body must repay its oxygen debt .
Type I vs. Type II Fibers
Several different types of muscle fiber can be found in most skeletal muscles: Type I and and 3 subtypes of Type II fibers. Each type differs in the myosin it uses and also in its structure and biochemistry.
Type I Fibers
- loaded with mitochondria
- depend on cellular respiration for ATP production
- fatty acids the major energy source
- resistant to fatigue
- rich in myoglobin and hence red in color (the "dark" meat of the turkey)
- activated by small-diameter, thus slow-conducting, motor neurons
- also known as "slow-twitch" fibers
- dominant in muscles used in activities requiring endurance (leg muscles) and those that depend on tonus, e.g., those responsible for posture
Type IIb Fibers
- few mitochondria
- rich in glycogen
- depend on creatine phosphate and glycolysis for ATP production
- fatigue easily with the production of lactic acid
- low in myoglobin hence whitish in color (the white meat of the turkey)
- activated by large-diameter, thus fast-conducting, motor neurons
- also known as "fast-twitch" fibers
- dominant in muscles used for rapid movement, e.g., those moving the eyeballs.
The other subtypes of Type II fibers have properties intermediate between those of Type IIb and Type I.
Most skeletal muscles contain some mixture of Type I and Type II fibers, but a single motor unit always contains one type or the other, never both.
In mice, the number of Type I vs Type II fibers can be changed with exercise and drug treatment. Whether the same holds true for humans remains to be seen. Perhaps training in humans does not alter the number of fibers of a particular type but may increase the diameter of one type (e.g., Type I in marathoners, Type IIb in weight lifters) at the expense of the other types.
Cardiac Muscle
Cardiac or heart muscle resembles skeletal muscle in some ways: it is striated and each cell contains sarcomeres with sliding filaments of actin and myosin. However, cardiac muscle has a number of unique features that reflect its function of pumping blood.
- The myofibrils of each cell (and cardiac muscle is made of single cells — each with a single nucleus) are branched.
- The branches interlock with those of adjacent fibers by adherens junctions. These strong junctions enable the heart to contract forcefully without ripping the fibers apart.
This electron micrograph (reproduced with permission from Keith R. Porter and Mary A. Bonneville, An Introduction to the Fine Structure of Cells and Tissues , 4th ed., Lea & Febiger, Philadelphia, 1973) shows an adherens junction and several of the other features listed here.
- The action potential that triggers the heartbeat is generated within the heart itself. Motor nerves (of the autonomic nervous system) do run to the heart, but their effect is simply to modulate — increase or decrease — the intrinsic rate and the strength of the heartbeat. Even if the nerves are destroyed (as they are in a transplanted heart), the heart continues to beat.
-
The action potential that drives contraction of the heart passes from fiber to fiber through gap junctions.
- Significance: all the fibers contract in a synchronous wave that sweeps from the atria down through the ventricles and pumps blood out of the heart. Anything that interferes with this synchronous wave (such as damage to part of the heart muscle from a heart attack) may cause the fibers of the heart to beat at random — called fibrillation . Fibrillation is the ultimate cause of most deaths, and its reversal is the function of defibrillators that are part of the equipment in ambulances, hospital emergency rooms, and even on U.S. air lines.
- The refractory period in heart muscle is longer than the period it takes for the muscle to contract (systole) and relax (diastole). Thus tetanus is not possible (a good thing, too!).
- Cardiac muscle has a much richer supply of mitochondria than skeletal muscle. This reflects its greater dependence on cellular respiration for ATP.
-
Cardiac muscle has little glycogen and gets little benefit from glycolysis when the supply of oxygen is limited.
- Thus anything that interrupts the flow of oxygenated blood to the heart leads quickly to damage - even death - of the affected part. This is what happens in heart attacks.
Smooth Muscle
Smooth muscle is made of single, spindle-shaped cells. It gets its name because no striations are visible in them. Nonetheless, each smooth muscle cell contains thick (myosin) and thin (actin) filaments that slide against each other to produce contraction of the cell. The thick and thin filaments are anchored near the plasma membrane (with the help of intermediate filaments).
Smooth muscle (like cardiac muscle) does not depend on motor neurons to be stimulated. However, motor neurons (of the autonomic system) reach smooth muscle and can stimulate it or relax it depending on the neurotransmitter they release (e.g. noradrenaline or nitric oxide, NO).
Smooth muscle can also be made to contract
-
by other substances released in the vicinity (paracrine stimulation)
- Example: release of histamine causes contraction of the smooth muscle lining our air passages (triggering an attack of asthma )
-
by hormones circulating in the blood
- Example: oxytocin reaching the uterus stimulates it to contract to begin childbirth.
The contraction of smooth muscle tends to be slower than that of striated muscle. It also is often sustained for long periods. This, too, is called tonus but the mechanism is not like that in skeletal muscle.
Muscle Diseases
The Muscular Dystrophies (MD)
Together myosin, actin, tropomyosin, and troponin make up over three-quarters of the protein in muscle fibers. Some two dozen other proteins make up the rest. These serve such functions as attaching and organizing the filaments in the sarcomere and connecting the sarcomeres to the plasma membrane and the extracellular matrix. Mutations in the genes encoding these proteins may produce defective proteins and resulting defects in the muscles.
Among the most common of the muscular dystrophies are those caused by mutations in the gene for dystrophin . The gene for dystrophin is huge, containing 79 exons spread out over 2.4 million base pairs of DNA. Thus this single gene represents about 0.1% of the entire human genome (3 x 10 9 bp) and is almost half the size of the entire genome of E. coli!
-
Duchenne muscular dystrophy
(
DMD
)
Deletions or nonsense mutations that cause a frameshift usually introduce premature termination codons (PTCs) in the resulting mRNA. Thus at best only a fragment of dystrophin is synthesized and DMD, a very severe form of the disease, results.
-
Becker muscular dystrophy
(
BMD
).
If the deletion simply removes certain exons but preserves the correct reading frame, a slightly-shortened protein results that produces BMD, a milder form of the disease.
The gene for dystrophin is on the X chromosome, so these two diseases strike males in a typical X-linked pattern of inheritance.
A treatment for DMD
Deletions of one or more exons in the huge dystrophin gene are the cause of most of the cases of DMD. Exon 50 is a particularly notorious offender. When it is deleted, splicing of the pre-mRNA introduces a frameshift which then introduces a premature termination codon resulting in no functional dystrophin synthesized ("
B
"). However, an antisense oligonucleotide targeted to exon 51 causes the splicing mechanism to skip over it resulting in the stitching together of exons 49 and 52. This restores the correct reading frame so that only a slightly-altered version of dystrophin is produced, i.e., a BMD-type dystrophin ("
C
"). Seventeen weeks of weekly injections of 12 young DMD patients in the Netherlands with the oligonucleotide caused their muscles to synthesize sufficient amounts of dystrophin to enable 8 of them to walk better than before. (See Goemans, N., et al., in the 21 April 2011 issue of
The New England Journal of Medicine
.
Three research groups have used the CRISPR-Cas9 genome editing system to remove a mutated exon in DMD mice. The treatment restored dystrophin synthesis and improved skeletal and cardiac muscle function in the mice.
Myasthenia Gravis
Myasthenia gravis is an autoimmune disorder affecting the neuromuscular junction. Patients have smaller end plate potentials (EPPs) than normal. With repeated stimulation, the EPPs become too small to trigger further action potentials and the fiber ceases to contract. Administration of an inhibitor of acetylcholinesterase temporarily can restore contractility by allowing more ACh to remain at the site.
Patients with myasthenia gravis have only 20% or so of the number of ACh receptors found in normal neuromuscular junctions. This loss appears to be caused by antibodies directed against the receptors. Some evidence:
- A disease resembling myasthenia gravis can be induced in experimental animals by immunizing them with purified ACh receptors.
- Anti-ACh receptor antibodies are found in the serum of human patients.
- Experimental animals injected with serum from human patients develop the signs of myasthenia gravis.
- Newborns of mothers with myasthenia gravis often show mild signs of the disease for a short time after their birth. This is the result of the transfer of the mother's antibodies across the placenta during gestation.
The reason some people develop autoimmune antibodies against the ACh receptor is unknown.
The Cardiac Myopathies
Cardiac muscle, like skeletal muscle, contains many proteins in addition to actin and myosin. Mutations in the genes for these may cause the wall of the heart to become weakened and, in due course, enlarged. Among the genes that have been implicated in these diseases are those encoding:
- actin
- two types of myosin
- troponin
- tropomyosin
- myosin-binding protein C (which links myosin to titin)
The severity of the disease varies with the particular mutation causing it (over 100 have been identified so far) . Some mutations are sufficiently dangerous that they can lead to sudden catastrophic heart failure in seemingly healthy and active young adults.


41.3: Excretion Systems
Skills to Develop
- Explain how vacuoles, present in microorganisms, work to excrete waste
- Describe the way in which flame cells and nephridia in worms perform excretory functions and maintain osmotic balance
- Explain how insects use Malpighian tubules to excrete wastes and maintain osmotic balance
Microorganisms and invertebrate animals use more primitive and simple mechanisms to get rid of their metabolic wastes than the mammalian system of kidney and urinary function. Three excretory systems evolved in organisms before complex kidneys: vacuoles, flame cells, and Malpighian tubules.
Contractile Vacuoles in Microorganisms
The most fundamental feature of life is the presence of a cell. In other words, a cell is the simplest functional unit of a life. Bacteria are unicellular, prokaryotic organisms that have some of the least complex life processes in place; however, prokaryotes such as bacteria do not contain membrane-bound vacuoles. The cells of microorganisms like bacteria, protozoa, and fungi are bound by cell membranes and use them to interact with the environment. Some cells, including some leucocytes in humans, are able to engulf food by endocytosis—the formation of vesicles by involution of the cell membrane within the cells. The same vesicles are able to interact and exchange metabolites with the intracellular environment. In some unicellular eukaryotic organisms such as the amoeba, shown in Figure \(\PageIndex{1}\), cellular wastes and excess water are excreted by exocytosis, when the contractile vacuoles merge with the cell membrane and expel wastes into the environment. Contractile vacuoles (CV) should not be confused with vacuoles, which store food or water.
Flame Cells of Planaria and Nephridia of Worms
As multi-cellular systems evolved to have organ systems that divided the metabolic needs of the body, individual organs evolved to perform the excretory function. Planaria are flatworms that live in fresh water. Their excretory system consists of two tubules connected to a highly branched duct system. The cells in the tubules are called flame cells (or protonephridia ) because they have a cluster of cilia that looks like a flickering flame when viewed under the microscope, as illustrated in Figure \(\PageIndex{2}\)a. The cilia propel waste matter down the tubules and out of the body through excretory pores that open on the body surface; cilia also draw water from the interstitial fluid, allowing for filtration. Any valuable metabolites are recovered by reabsorption. Flame cells are found in flatworms, including parasitic tapeworms and free-living planaria. They also maintain the organism’s osmotic balance.
Earthworms (annelids) have slightly more evolved excretory structures called nephridia , illustrated in Figure \(\PageIndex{2}\)b. A pair of nephridia is present on each segment of the earthworm. They are similar to flame cells in that they have a tubule with cilia. Excretion occurs through a pore called the nephridiopore . They are more evolved than the flame cells in that they have a system for tubular reabsorption by a capillary network before excretion.
Malpighian Tubules of Insects
Malpighian tubules are found lining the gut of some species of arthropods, such as the bee illustrated in Figure \(\PageIndex{3}\). They are usually found in pairs and the number of tubules varies with the species of insect. Malpighian tubules are convoluted, which increases their surface area, and they are lined with microvilli for reabsorption and maintenance of osmotic balance. Malpighian tubules work cooperatively with specialized glands in the wall of the rectum. Body fluids are not filtered as in the case of nephridia; urine is produced by tubular secretion mechanisms by the cells lining the Malpighian tubules that are bathed in hemolymph (a mixture of blood and interstitial fluid that is found in insects and other arthropods as well as most mollusks). Metabolic wastes like uric acid freely diffuse into the tubules. There are exchange pumps lining the tubules, which actively transport H + ions into the cell and K + or Na + ions out; water passively follows to form urine. The secretion of ions alters the osmotic pressure which draws water, electrolytes, and nitrogenous waste (uric acid) into the tubules. Water and electrolytes are reabsorbed when these organisms are faced with low-water environments, and uric acid is excreted as a thick paste or powder. Not dissolving wastes in water helps these organisms to conserve water; this is especially important for life in dry environments.
Link to Learning
Visit this site to see a dissected cockroach, including a close-up look at its Malpighian tubules.
Summary
Many systems have evolved for excreting wastes that are simpler than the kidney and urinary systems of vertebrate animals. The simplest system is that of contractile vacuoles present in microorganisms. Flame cells and nephridia in worms perform excretory functions and maintain osmotic balance. Some insects have evolved Malpighian tubules to excrete wastes and maintain osmotic balance.
Glossary
- flame cell
- (also, protonephridia) excretory cell found in flatworms
- Malpighian tubule
- excretory tubules found in arthropods
- microvilli
- cellular processes that increase the surface area of cells
- nephridia
- excretory structures found in annelids
- nephridiopore
- pore found at the end of nephridia


9.1.2: Costs of Running a Restaurant
Lesson
Let's explore how much the food will cost.
Exercise \(\PageIndex{1}\): Introducing Spreadsheets
-
Type each formula into the cells of a spreadsheet program and press enter. Record what the cell displays. Make sure to type each formula exactly as it is written here.
A B C D 1 \(=40-32\) \(=1.5+3.6\) \(=14/7\) \(=0.5*6\) Table \(\PageIndex{1}\) -
- Predict what will happen if you type the formula =A1*C1 into cell C2 of your spreadsheet.
- Type in the formula, and press enter to check your prediction.
-
- Predict what will happen next if you delete the formula in cell A1 and replace it with the number 100.
- Replace the formula with the number, and press enter to check your prediction.
-
- Predict what will happen if you copy cell C2 and paste it into cell D2 of your spreadsheet.
- Copy and paste the formula to check your prediction.
Exercise \(\PageIndex{2}\): Cost per Serving
-
Set up a spreadsheet with these column labels in the first row.
Table \(\PageIndex{2}\)A B C D 1 ingredient unit in recipe amount in recipe amount per serving 2
- Type the information about the ingredients in your recipe into the first 3 columns of the spreadsheet.
- Type a formula into cell D2 to automatically calculate the amount per serving for your first ingredient.
- Copy cell D2 and paste it into the cells beneath it to calculate the amount per serving for the rest of your ingredients. Pause here so your teacher can review your work.
-
Add these column labels to your spreadsheet.
Table \(\PageIndex{3}\)E F G H purchase price purchase amount purchase unit cost per purchase unit - Research the cost of each ingredient in your meal, and record the information in columns E, F, and G.
- Type a formula into cell H2 to automatically calculate the cost per purchase unit for your first ingredient.
- Copy cell H2, and paste it into the cells beneath it to calculate the cost per purchase unit for the rest of your ingredients.
-
Add these column labels to your spreadsheet.
Table \(\PageIndex{4}\)I J K conversion from purchase units to recipe units cost per recipe unit cost per serving - Complete column I with how many of your recipe unit are in 1 of your purchase unit for each ingredient. For example, if your recipe unit was cups and your purchase unit was gallons, then your conversion would be 16 because there are 16 cups in 1 gallon.
- Type a formula into cell J2 to calculate the cost per recipe unit for your first ingredient.
- Type a formula into cell K2 to calculate the cost per serving for your first ingredient.
- Compare formulas with your partner. Discuss your thinking. If you disagree, work to reach an agreement.
- Copy cells J2 and K2, and paste them into the cells beneath them to calculate the cost per recipe unit and cost per serving for the rest of your ingredients.
- Type a formula into the first empty cell below your last ingredient in column K to calculate the total cost per serving for all of the ingredients in your recipe. Record the answer here.


13.4: Outcome- Marketing Communication Methods
What you’ll learn to do: describe common methods of marketing communication, their advantages and disadvantages
A common challenge for people new to marketing is learning about the many marketing communication tools and methods now available and understanding how to use them effectively. Fortunately, most of us have first-hand experience of being on the receiving end of IMC—whether you like it or not, you are a consumer, and you’ve been the target of all kinds of marketing communication. You know the difference between an ad that gets your attention and one you just tune out, for example. You are familiar with the line between “persistent” and “annoying” when it comes to getting marketing-related emails or text messages. You recognize which buy-one-get-one-free offers are a great deal, and which ones seem like a racket.
All this experience will come in handy in this section and later in the course. You’re getting closer to being on the other side of the wall, where you’ll be tasked with using marketing communication methods and tools to devise your own marketing plan (in the last module of this course!). In this section, though, you’ll get a chance to examine each of these marketing communication methods one by one. Fortunately, the underlying principles we’ve discussed up to this point apply to all of them: knowing your audience, defining strategy, setting objectives, crafting the message. Where paths diverge is in the tools themselves: how to design a great ad, how to produce a memorable event, how to get coverage for your organization in the news media, how to use email and social media skillfully for marketing purposes, and so on.
The next several readings provide a general overview of seven important marketing communication methods (shown in Figure 1, below) in common use today. This section will help you become familiar with each method, common tools associated with each method, how to use these methods effectively, and the advantages and disadvantages of each one. Some marketing professionals spend entire careers becoming specialists in one or more of these areas. Other marketers become generalists who are skilled at bringing together different tools–and experts–to execute effective IMC programs.
Whether you think you’re more of a generalist or a specialist, marketing offers great opportunities for creativity and experimentation. There will always be a new idea, strategy, tool, or combination of tactics that marketers can turn into IMC magic for their companies and their customers. As you learn about and gain experience with the basic tools and approaches, you’ll see opportunities to try something new. And you should: in the marketing world, fresh is good!
The specific things you’ll learn in this section include:
- Explain Advertising
- Explain Public Relations
- Explain Sales Promotions
- Explain Personal Selling
- Explain Direct Marketing
- Explain Digital Marketing
- Explain Guerrilla Marketing
Learning Activities
- Reading: Advertising
- Reading: Public Relations
- Reading: Sales Promotions
- Reading: Personal Selling
- Reading: Direct Marketing
- Reading: Digital Marketing
- Reading: Guerrilla Marketing
- Simulation: Integrated Marketing
- Self Check: Marketing Communication Methods
Contributors and Attributions
- Outcome: Marketing Communication Methods. Provided by : Lumen Learning. License : CC BY: Attribution


11.5: Semiconductors- Band Gaps, Colors, Conductivity and Doping
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Semiconductors, as we noted above, are somewhat arbitrarily defined as insulators with band gap energy < 3.0 eV (~290 kJ/mol). This cutoff is chosen because, as we will see, the conductivity of undoped semiconductors drops off exponentially with the band gap energy and at 3.0 eV it is very low. Also, materials with wider band gaps (e.g. SrTiO 3 , E gap = 3.2 eV) do not absorb light in the visible part of the spectrum
There are a number of places where we find semiconductors in the periodic table:
- Early transition metal oxides and nitrides, especially those with d 0 electron counts such as TiO 2 , TaON, and WO 3
- Oxides of later 3d elements such as Fe 2 O 3 , NiO, and Cu 2 O
- Layered transition metal chalcogenides with d 0 , d 2 and d 6 electron counts including TiS 2 , ZrS 2 , MoS 2 , WSe 2 , and PtS 2
- d 10 copper and sliver halides, e.g., CuI, AgBr, and AgI
- Zincblende- and wurtzite-structure compounds of the p-block elements, especially those that are isoelectronic with Si or Ge, such as GaAs and CdTe. While these are most common, there are other p-block semiconductors that are not isoelectronic and have different structures, including GaS, PbS, and Se.
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A 2" wafer cut from a GaAs single crystal. GaAs, like many p-block semiconductors, has the zincblende structure. |
The p-block octet semiconductors are by far the most studied and important for technological applications, and are the ones that we will discuss in detail.
Zincblende- and wurtzite-structure semiconductors have 8 valence electrons per 2 atoms. These combinations include 4-4 (Si, Ge, SiC,…), 3-5 (GaAs, AlSb, InP,…), 2-6 (CdSe, HgTe, ZnO,…), and 1-7 (AgCl, CuBr,…) semiconductors. Other variations that add up to an octet configuration are also possible, such as Cu I In III Se 2 , which has the chalcopyrite structure, shown at the right.
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The chalcopyrite structure is adopted by ABX 2 octet semiconductors such as Cu I In III Se 2 and Cd II Sn IV P 2 . The unit cell is doubled relative to the parent zincblende structure because of the ordered arrangement of cations. Each anion (yellow) is coordinated by two cations of each type (blue and red). |
How does the band gap energy vary with composition ? There are two important trends
(1) Going down a group in the periodic table, the gap decreases :
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- C (diamond) > Si > Ge > α-Sn
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E gap (eV): 5.4 1.1 0.7 0.0
This trend can be understood by recalling that E gap is related to the energy splitting between bonding and antibonding orbitals . This difference decreases (and bonds become weaker) as the principal quantum number increases.
(2) For isoelectronic compounds, increasing ionicity results in a larger band gap .
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- Ge < GaAs < ZnSe
- 0.7 1.4 2.8 eV
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- Sn < InSb < CdTe < AgI
- 0.0 0.2 1.6 2.8 eV
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This trend can also be understood from a simple MO picture, as we discussed in Ch. 2. As the electronegativity difference Δχ increases, so does the energy difference between bonding and antibonding orbitals.
The band gap is a very important property of a semiconductor because it determines its color and conductivity. Many of the applications of semiconductors are related to band gaps:
- Narrow gap materials (Hg x Cd 1 - x Te, VO 2 , InSb, Bi 2 Te 3 ) are used as infrared photodetectors and thermoelectrics (which convert heat to electricity).
- Wider gap materials (Si, GaAs, GaP, GaN, CdTe, CuIn x Ga 1-x Se 2 ) are used in electronics, light-emitting diodes, and solar cells.
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Color wheel showing the colors and wavelengths of emitted light. |
Semiconductor solid solutions such as GaAs 1-x P x have band gaps that are intermediate between the end member compounds, in this case GaAs and GaP (both zincblende structure). Often, there is a linear relation between composition and band gap, which is referred to as Vegard's Law . This "law" is often violated in real materials, but nevertheless offers useful guidance for designing materials with specific band gaps. For example, red and orange light-emitting diodes (LED's) are made from solid solutions with compositions of GaP 0.40 As 0.60 and GaP 0.65 As 0.35 , respectively. Increasing the mole fraction of the lighter element (P) results in a larger band gap, and thus a higher energy of emitted photons.
Colors of semiconductors
The color of absorbed and emitted light both depend on the band gap of the semiconductor. Visible light covers the range of approximately 390-700 nm, or 1.8-3.1 eV. The color of emitted light from an LED or semiconductor laser corresponds to the band gap energy and can be read off the color wheel shown at the right.
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Fe 2 O 3 powder is reddish orange because of its 2.2 eV band gap |
The color of absorbed light includes the band gap energy, but also all colors of higher energy (shorter wavelength), because electrons can be excited from the valence band to a range of energies in the conduction band. Thus semiconductors with band gaps in the infrared (e.g., Si, 1.1 eV and GaAs, 1.4 eV) appear black because they absorb all colors of visible light. Wide band gap semiconductors such as TiO 2 (3.0 eV) are white because they absorb only in the UV. Fe 2 O 3 has a band gap of 2.2 eV and thus absorbs light with λ < 560 nm. It thus appears reddish-orange (the colors of light reflected from Fe 2 O 3 ) because it absorbs green, blue, and violet light. Similarly, CdS (E gap = 2.6 eV) is yellow because it absorbs blue and violet light.
Electrons and holes in semiconductors
Pure (undoped) semiconductors can conduct electricity when electrons are promoted, either by heat or light, from the valence band to the conduction band. The promotion of an electron (e - ) leaves behind a hole (h + ) in the valence band. The hole, which is the absence of an electron in a bonding orbital, is also a mobile charge carrier, but with a positive charge. The motion of holes in the lattice can be pictured as analogous to the movement of an empty seat in a crowded theater. An empty seat in the middle of a row can move to the end of the row (to accommodate a person arriving late to the movie) if everyone moves over by one seat. Because the movement of the hole is in the opposite direction of electron movement, it acts as a positive charge carrier in an electric field.
The opposite process of excitation, which creates an electron-hole pair, is their recombination. When a conduction band electron drops down to recombine with a valence band hole, both are annihilated and energy is released. This release of energy is responsible for the emission of light in LEDs.
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An electron-hole pair is created by adding heat or light energy E > E gap to a semiconductor (blue arrow). The electron-hole pair recombines to release energy equal to E gap (red arrow). |
At equilibrium, the creation and annihilation of electron-hole pairs proceed at equal rates. This dynamic equilibrium is analogous to the dissociation-association equilibrium of H + and OH - ions in water. We can write a mass action expression:
\(n \times p = K_{eq} = n_{i}^{2}\)
where n and p represent the number density of electrons and holes, respectively, in units of cm -3 . The intrinsic carrier concentration, n i , is equal to the number density of electrons or holes in an undoped semiconductor, where n = p = n i .
Note the similarity to the equation for water autodissociation:
\([H^{+}][OH^{-}] = K_{w}\)
By analogy, we will see that when we increase n (e.g., by doping), p will decrease, and vice-versa, but their product will remain constant at a given temperature.
Temperature dependence of the carrier concentration. Using the equations \(K_{eq} = e^{(\frac{- \Delta G^{o}}{RT})} \) and \(\Delta G^{o} = \Delta H^{o} - T \Delta S^{o}\), we can write:
\[ n \times p = n_{i}^{2} = e^{(\frac{\Delta S^{o}} {R})} e^{(\frac{- \Delta H^{o}}{RT})}\]
The entropy change for creating electron hole pairs is given by:
\[\Delta S^{o} = R ln (N_{V}) + R ln (N_{V}) = R ln (N_{C}N_{V})\]
where N V and N C are the effective density of states in the valence and conduction bands, respectively.
and thus we obtain
\[n_{i}^{2} = N_{C}N_{V} e^{({- \Delta H^{o}}{RT})}\]
Since the volume change is negligible, \(\Delta H^{o} \approx \Delta E^{o}\), and therefore \(\frac {\Delta H^{o}}{R} \approx \frac{E_{gap}}{k}\), from which we obtain
\[n_{i}^{2} = N_{C}N_{V} e^{(\frac{-E_{gap}}{kT})}\]
and finally
\[\mathbf{n= p = n_{i} = (N_{C}N_{V})^{\frac{1}{2}} e^{(\frac{-E_{gap}}{2kT})}}\]
For pure Si (E gap = 1.1 eV) with N ≈ 10 22 /cm 3 , we can calculate from this equation a carrier density n i of approximately 10 10 /cm 3 at 300 K. This is about 12 orders of magnitude lower than the valence electron density of Al, the element just to the left of Si in the periodic table. Thus we expect the conductivity of pure semiconductors to be many orders of magnitude lower than those of metals.
Conductivity of intrinsic semiconductors
The conductivity (σ) is the product of the number density of carriers (n or p), their charge (e), and their mobility (µ). Recall from Chapter 6 that µ is the ratio of the carrier drift velocity to the electric field and has units of cm 2 /Volt-second. Typically electrons and holes have somewhat different mobilities (µ e and µ h , respectively) so the conductivity is given by:
\[\sigma = ne\mu_{e} + pe\mu_{h}\]
For either type of charge carrier, we recall from Ch. 6 that the mobility μ is given by:
\[\mu = \frac{v_{drift}}{E} = \frac{e\tau}{m}\]
where e is the fundamental unit of charge, τ is the scattering time, and m is the effective mass of the charge carrier.
Taking an average of the electron and hole mobilities, and using n = p, we obtain
\[\mathbf{\sigma= \sigma_{o} e^{(\frac{-E_{gap}}{2kT})}}, \: where \: \sigma_{o} = 2(N_{C}N_{V})^{\frac{1}{2}}e\mu\]
By measuring the conductivity as a function of temperature, it is possible to obtain the activation energy for conduction, which is E gap /2. This kind of plot, which resembles an Arrhenius plot, is shown at the right for three different undoped semiconductors. The slope of the line in each case is -E gap /2k.
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Plots of ln(σ) vs. inverse temperature for intrinsic semiconductors Ge (E gap = 0.7 eV), Si (1.1 eV) and GaAs (1.4 eV). The slope of the line is -E gap /2k. |
Doping of semiconductors. Almost all applications of semiconductors involve controlled doping , which is the substitution of impurity atoms, into the lattice. Very small amounts of dopants (in the parts-per-million range) dramatically affect the conductivity of semiconductors. For this reason, very pure semiconductor materials that are carefully doped - both in terms of the concentration and spatial distribution of impurity atoms - are needed.
n- and p-type doping . In crystalline Si, each atom has four valence electrons and makes four bonds to its neighbors. This is exactly the right number of electrons to completely fill the valence band of the semiconductor. Introducing a phosphorus atom into the lattice (the positively charged atom in the figure at the right) adds an extra electron, because P has five valence electrons and only needs four to make bonds to its neighbors. The extra electron, at low temperature, is bound to the phosphorus atom in a hydrogen-like molecular orbital that is much larger than the 3s orbital of an isolated P atom because of the high dielectric constant of the semiconductor. In silicon, this "expanded" Bohr radius is about 42 Å, i.e., 80 times larger than in the hydrogen atom. The energy needed to ionize this electron – to allow it to move freely in the lattice - is only about 40–50 meV, which is not much larger the thermal energy (26 meV) at room temperature. Therefore the Fermi level lies just below the conduction band edge, and a large fraction of these extra electrons are promoted to the conduction band at room temperature, leaving behind fixed positive charges on the P atom sites. The crystal is n-doped , meaning that the majority carrier (electron) is n egatively charged.
Alternatively, boron can be substituted for silicon in the lattice, resulting in p-type doping, in which the majority carrier (hole) is p ositively charged. Boron has only three valence electrons, and "borrows" one from the Si lattice, creating a positively charged hole that exists in a large hydrogen-like orbital around the B atom. This hole can become delocalized by promoting an electron from the valence band to fill the localized hole state. Again, this process requires only 40–50 meV, and so at room temperature a large fraction of the holes introduced by boron doping exist in delocalized valence band states. The Fermi level (the electron energy level that has a 50% probability of occupancy at zero temperature) lies just above the valence band edge in a p-type semiconductor.
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n- and p-type doping of semiconductors involves substitution of electron donor atoms (light orange) or acceptor atoms (blue) into the lattice. These substitutions introduce extra electrons or holes, respectively, which are easily ionized by thermal energy to become free carriers. The Fermi level of a doped semiconductor is a few tens of mV below the conduction band (n-type) or above the valence band (p-type). |
As noted above, the doping of semiconductors dramatically changes their conductivity. For example, the intrinsic carrier concentration in Si at 300 K is about 10 10 cm -3 . The mass action equilibrium for electrons and holes also applies to doped semiconductors, so we can write:
\[n \times p = n_{i}^{2} = 10^{20} cm^{-6} \: at \: 300K\]
If we substitute P for Si at the level of one part-per-million, the concentration of electrons is about 10 16 cm -3 , since there are approximately 10 22 Si atoms/cm 3 in the crystal. According to the mass action equation, if n = 10 16 , then p = 10 4 cm -3 . There are three consequences of this calculation:
- The density of carriers in the doped semiconductor (10 16 cm -3 ) is much higher than in the undoped material (~10 10 cm -3 ), so the conductivity is also many orders of magnitude higher.
- The activation energy for conduction is only 40–50 meV, so the conductivity does not change much with temperature (unlike in the intrinsic semiconductor)
- The minority carriers (in this case holes) do not contribute to the conductivity, because their concentration is so much lower than that of the majority carrier (electrons).
Similarly, for p-type materials, the conductivity is dominated by holes, and is also much higher than that of the intrinsic semiconductor.
Chemistry of semiconductor doping. Sometimes it is not immediately obvious what kind of doping (n- or p-type) is induced by "messing up" a semiconductor crystal lattice. In addition to substitution of impurity atoms on normal lattice sites (the examples given above for Si), it is also possible to dope with vacancies - missing atoms - and with interstitials - extra atoms on sites that are not ordinarily occupied. Some simple rules are as follows:
- For substitutions, adding an atom to the right in the periodic table results in n-type doping, and an atom to the left in p-type doping.
For example, when TiO 2 is doped with Nb on some of the Ti sites, or with F on O sites, the result is n-type doping. In both cases, the impurity atom has one more valence electron than the atom for which it was substituted. Similarly, substituting a small amount of Zn for Ga in GaAs, or a small amount of Li for Ni in NiO, results in p-type doping.
- Anion vacancies result in n-type doping, and cation vacancies in p-type doping.
Examples are anion vacancies in CdS 1-x and WO 3-x , both of which give n-type semiconductors, and copper vacancies in Cu 1-x O, which gives a p-type semiconductor.
- Interstitial cations (e.g. Li) donate electrons to the lattice resulting in n-type doping. Interstitial anions are rather rare but would result in p-type doping.
Sometimes, there can be both p- and n-type dopants in the same crystal, for example B and P impurities in a Si lattice, or cation and anion vacancies in a metal oxide lattice. In this case, the two kinds of doping compensate each other, and the doping type is determined by the one that is in higher concentration. A dopant can also be present on more than one site. For example, Si can occupy both the Ga and As sites in GaAs, and the two substitutions compensate each other. Si has a slight preference for the Ga site, however, resulting in n-type doping.


12.2: The Principles of Articulation
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When considering how to articulate a course for transfer credit, evaluators are faced with numerous decisions. Fortunately, they can turn to a number of principles to guide them as they try to ensure that courses are articulated fairly and consistently. These can be divided into foundational principles, operating principles, and provisional principles.
Foundational Principles
Foundational principles are those which lie at the core of decisions about all articulation of courses and programs.
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Equivalence: Equivalent means “equal in value”. A course submitted for articulation will likely never be identical to the corresponding course at the receiving institution. The assessment of equivalence involves identifying the degree to which it matches in content or outcomes. Discipline and program contexts will dictate the relative importance of the similarity.
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In lieu: The act of awarding transfer credit implies the acceptance of a course in place of a course or program requirement offered at the receiving institution. The course to be transferred does not have to be identical to the course for which transfer credit is granted, but the degree of similarity should ensure that students will have the necessary knowledge and background to be successful in more advanced courses.
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Applicability: It is appropriate to award transfer credit for courses that can be used to fulfill the specific or general requirements of a credential or program at the receiving institution.
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Fairness: Provisos and restrictions (such as adding a specific grade requirement) should not be placed on equivalent courses unless those same restrictions apply at the institution awarding the transfer credit, or there are clear and defensible reasons for doing so.
Situational Principles
Situational principles provide useful guidance but are not universally applicable. While they form part of the decision-making toolkit for articulation, situations and contexts create provisos for their application. Two such principles are relevant to the articulation of online courses.
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Pedagogy: Under some circumstances it is appropriate to consider how a course is taught. Factors such as cultural sensitivity, or opportunities for practising skills, may be integral to content mastery. See “Awarding Credit” below, for more on pedagogy.
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Delivery: How a course is delivered is normally immaterial to its articulation, since teaching a course in a distance delivery format (as opposed to face-to-face) should not affect its equivalence. However, there may be occasions where the content is intrinsically linked to delivery, and an alternative mode impacts on equivalence. It may also be relevant whether a course is offered only online, or if an online course is a version of a course normally delivered in a traditional classroom.
Operational Principles
Operational principles refer to practices and attitudes that will facilitate articulation. In the case of online courses the following two are relevant:
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Comparability: Since it should be possible to compare courses, the elements of the course must be clearly outlined and should be interpretable by faculty in the same or a related field. The best assurance of comparability is a course outline that is comprehensive enough to allow for the assessment of equivalence, and that conforms broadly or specifically to the local norms of course description.
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Transparency: Assessment practices should be open to scrutiny. Any individual who assigns transfer credit based on the assessment of a course should be prepared to explain the reasons for the decision, including any influencing factors.
Using The Principles to Request and Assess Credit for Online Courses
The course developer (at the sending institution) and the course assessor (at the receiving institution) both have a part to play in ensuring that appropriate transfer credit will be allocated when a student transfers. The onus is on the course developer to provide accurate, detailed and honest information about the course, while the assessor must base his or her decision on sound principles, and act fairly and in the best interests of the student.


5.2: Definitions
The Importance of Nonverbal Communication
Nonverbal communication plays a significant role in our daily interactions with others. As you can see in Figure \(\PageIndex{1}\), there is a lot of nonverbal expression on a baby’s face. How directly or indirectly we look at someone communicates a message. Speaking loudly or softly can change the meaning of a phrase. A touch may send a comforting message, make a connection with someone, or even serve as a display of power. Smells may remind us of a favorite holiday (such as the smell of vanilla) or encourage us to relax (such as the smell of lavender). Even taste might communicate a message. For example, a root beer float on a hot, sunny day could bring back good memories of childhood.
Nonverbals [sic] is everything that communicates but is not a word. This beautiful theater, it's communicating to us.  The things that you attach to yourself—a purse, a pen, a fancy car—all these things are communicating. How you look at others communicates. And all day long, we are communicating nonverbally. All day long. (TED, 2020)
This excerpt is from the transcript for the TEDxManchester talk titled “ The Power of Nonverbal Communication ,” delivered by Joe Navarro. Who is Navarro and why consider what he has to say about nonverbal communication? Navarro has spent his career using and sharing his knowledge of nonverbal communication:
For 25 years, Joe used nonverbal communication as his primary professional tool in the FBI to catch and interview spies, criminals, and terrorists. Since leaving the FBI his life’s work has been in building awareness about the power of nonverbal communication, debunking myths and falsehoods about nonverbals and giving people insight in how to better utilize nonverbals to enhance their communication skills. (TED, 2020)
It’s just not professionals working for the FBI who care about nonverbal communication. A search of TED Talks on the topic of nonverbal communication came up with a list of 11 videos.
So what is nonverbal communication? Why is it such a popular topic?
Defining Nonverbal Communication
According to the APA Dictionary of Psychology (n.d.), nonverbal communication is “the act of conveying information without the use of words. Nonverbal communication occurs through facial expressions, gestures, body language, tone of voice, and other physical indications of mood, attitude, approbation, and so forth, some of which may require knowledge of the culture or subculture to understand.” That definition covers a lot of factors!
Many scholars use the term nonverbal communication to designate those communication cues that express ideas and the meanings of our words (our “verbal communication”). Matsumoto et. al (2012) define “nonverbal communication as the transfer and exchange of messages in any and all modalities that do not involve words.” Hall and Knapp (2013) identify “all potentially informative behaviors that are not purely linguistic in content” as nonverbal cues.”
When we introduce nonverbal communication in our Communication courses, we make the distinction between verbal communication (the words) and nonverbal communication (everything else!). That even includes how we say the words. We will discuss this in more detail later on in the chapter, but here is an example for you to think about:
I see that you have a new backpack. I say “Wow, that’s a great backpack.” I could say this in a positive way with my facial expression and my voice. I could say the same words with a “skeptical” look on my face in a sarcastic tone of voice. I am using the same set of words, but, with a different nonverbal delivery of those words, I am sending you a different message.
For the purposes of this textbook, we define nonverbal communication as everything that conveys meaning beyond the words themselves. Nonverbal communication includes the use of all five senses—seeing, hearing, touching, tasting, and smelling—in creating meaning by the speaker/writer and how that meaning is interpreted by the listener/reader.
Many of these nonverbal cues may be visible or auditory. According to Hall and Knapp (2013):
Visible nonverbal cues include facial expressions, head movements, posture, body and hand movements, self and other-touching, leg positions and movements, interpersonal gaze, directness of interpersonal orientation, interpersonal distance, and synchrony or mimicry between people. Auditory nonverbal cues include discrete nonlinguistic vocal sounds (e.g., sighs) as well as qualities of the voice such as pitch and pitch variation, loudness, speed and speed variation, and tonal qualities. This list does not exhaust the many kinds of nonverbal communication that are present in the human behavioral repertoire.
As stated earlier, even how we say words—vocally or in print—communicate nonverbal messages. For example, we could say something in a way that shows respect, or we could say the same thing in a disconfirming or even angry tone of voice. We could also do this in print by the USE OF CAPITALS, which suggest that we are shouting at you. We will talk more about this later on in the chapter.
Figure \(\PageIndex{2}\) shows an emoji “thinking” by a hand is placed on the chin while the eyebrows are pointing down. We can look at this image and interpret those nonverbal cues as communicating that little yellow face as “deep in thought.” So much of our communication is now “computer-mediated communication.” We are all familiar with the use of :-) (smiley face) and the :-( (unhappy face). According to the definition in Encyclopedia Britannica (2022), an emoticon "is meant to represent a facial expression in order to communicate the emotional state of the author…. The word emoticon comes from a combination of the words emotion and icon ." Figure \(\PageIndex{3}\) shows different of emojis.
During the pandemic, wore a face mask, which had an impact on our nonverbal communication. Even emojis joined in on this trend. What do the emojis wearing face masks in Figure \(\PageIndex{4}\) represent to you? What do they communicate?
Here are a few more questions to think about: Are emoticons and emojis nonverbal communication, verbal communication, or both? Are they almost a “universal language” that transcends verbal languages because of their common nonverbal meanings? What cultural differences are there in the use of these images? What do you think?


15.24A: Sepsis and Septic Shock
- Compare and contrast the symptoms of: sepsis, severe sepsis, septic shock
Sepsis is a potentially deadly medical condition characterized by a whole-body inflammatory state (called a systemic inflammatory response syndrome or SIRS) that is triggered by an infection. Septic shock is a medical condition as a result of severe infection and sepsis, though the microbe may be systemic or localized to a particular site. Its most common victims are children, immuno-compromised individuals, and the elderly, as their immune systems cannot deal with the infection as effectively as those of healthy adults. Frequently, patients suffering from septic shock are cared for in intensive care units. The mortality rate from septic shock is approximately 25–50%.
Sepsis
Sepsis is an illness in which the body has a severe response to bacteria or other germs. The body may develop this inflammatory response by the immune system to microbes in the blood, urine, lungs, skin, or other tissues. A popular term for sepsis is blood poisoning. Severe sepsis is the systemic inflammatory response, infection, and the presence of organ dysfunction.
A bacterial infection anywhere in the body may set off the response that leads to sepsis. Common places where an infection might start include:
- the bloodstream
- bones (common in children)
- the bowel (usually seen with peritonitis)
- the kidneys (upper urinary tract infection or pyelonephritis )
- the lining of the brain ( meningitis )
- the liver or gallbladder
- the lungs (bacterial pneumonia )
- the skin (cellulitis)
For patients in the hospital, common sites of infection include intravenous lines, surgical wounds, surgical drains, and sites of skin breakdown known as bedsores (decubitus ulcers).
The therapy of sepsis rests on intravenous fluids, antibiotics, surgical drainage of infected fluid collections, and appropriate support for organ dysfunction. This may include hemodialysis in kidney failure, mechanical ventilation in pulmonary dysfunction, transfusion of blood products, and drug and fluid therapy for circulatory failure. Ensuring adequate nutrition—preferably by enteral feeding, but if necessary by parenteral nutrition—is important during prolonged illness.
Septic Shock
In sepsis, blood pressure drops, resulting in septic shock. Major organs and body systems, including the kidneys, liver, lungs, and central nervous system, stop working properly because of poor blood flow.
Most cases of septic shock are caused by Gram-positive bacteria, followed by endotoxin-producing Gram-negative bacteria. Endotoxins are bacterial membrane lipopolysaccharides (LPS) consisting of a toxic fatty acid (lipid A) core common to all Gram-negative bacteria, and a complex polysaccharide coat (including O antigen) unique for each species. Analogous molecules in the walls of Gram-positive bacteria and fungi can also elicit septic shock. In Gram-negative sepsis, free LPS attaches to a circulating LPS-binding protein, and the complex then binds to a specific receptor (CD14) on monocytes, macrophages, and neutrophils.
If sepsis worsens to the point of end-organ dysfunction (renal failure, liver dysfunction, altered mental status, or heart damage) then the condition is called severe sepsis. Once severe sepsis worsens to the point where blood pressure can no longer be maintained with intravenous fluids alone, then the criteria have been met for septic shock. The precipitating infections which may lead to septic shock if severe enough include appendicitis, pneumonia, bacteremia, diverticulitis, pyelonephritis, meningitis, pancreatitis, and necrotizing fasciitis.
Treatment primarily consists of the following:
- Volume resuscitation
- Early antibiotic administration
- Early goal directed thearpy
- Rapid source identification and control.
- Support of major organ dysfunction.
There are new drugs that act against the extreme inflammatory response seen in septic shock. These may help limit organ damage.
The mortality rate from sepsis is approximately 40% in adults, and 25% in children, and is significantly greater when left untreated for more than seven days.
Key Points
- Sepsis results from certain bacterial infections, often acquired in a hospital. Having certain conditions, such as a weakened immune system, certain chronic disorders, an artificial joint, or heart valve increases the risk.
- Symptoms of sepsis include either fever or low body temperature, rapid breathing, chills and shaking, rapid heartbeat, decreased urine output, and confusion or delirium.
- Severe sepsis often causes extremely low blood pressure, which limits blood flow to the body and can result in organ failure and death. This is known as septic shock.
- Sepsis is treated with antibiotics, fluids, and medicines to support blood pressure and prevent organ damage.
Key Terms
- septic shock : A life-threatening condition caused by infection and sepsis, often after surgery or trauma.
- sepsis : A life-threatening medical condition caused by a severe inflammatory response of the human body triggered by the presence of an infectious agent.
- mortality rate : the number of deaths per given unit of population over a given period of time


7.3: Reading- Security and Social Issues
Security Within the Organization
Security risks within an organization include the processing of fraudulent transactions, unauthorized access to data and program files, and the physical theft or damage of equipment.
Fraud
Computer fraud is increasing at an alarming rate. Fraud can be defined as the manipulation of the records of an organization to conceal an illegal act (normally the theft of funds or other assets). Computers can make it easy for employees in particular to defraud the organization, in particular when the level of security and internal control is lax. In manual systems, a common control to limit fraud is to involve two or more people in a process, each one effectively controlling the activities of the others. We call this control process separation of duties. For example in a payroll system one might give an individual the authority to approve increases, another the task of updating the computer and the third the responsibility to distribute funds to employees. Without collusion between them, it would be difficult for any one of these individuals to steal funds from the payroll system and hide his tracks. Unfortunately, in many computer systems, too many separate functions have been computerized and often there is a single clerk responsible for running the entire payroll process. In these situations, anyone who has access to the application can take the opportunity to commit fraud. The most common fraud tactics are:
- Entering fictitious transactions. Most frauds are committed by employees using the system in the normal way to enter fictitious transactions. No special technical knowledge is required and the employee relies on the fact that management supervision of the process in weak.
- Modification of computer files. Normally requires a little more technical expertise as this would involve, for example, the increase or reduction of amounts held on the master file, which cannot be changed within the application without an appropriate transaction (such as a payment).
- Unauthorized changes to programs. This type of fraud is usually limited to staff with programming expertise. A common example is the skimming or salami technique. In a payroll system this would entail deducting a small amount from each individual salary cheque and adding the total to a select individual’s payment. The secret behind this technique is that employees are unlikely to notice a change in their salary (PAYE and other deductions often cause regular variation in the total) and the total payroll will balance (the total amount being paid is the same.)
How does an organization limit fraud? Experts suggest a three-pronged attack. Firstly the organization must stress the need for honesty and ethical behavior in all business activities. Managers must lead by example, new employees must be screened and staff training must support this theme.  There must be strong internal controls, separation of duties, restricted access to sensitive applications and constant management supervision. Audit trails are used to record the origin of every transaction, and sequential numbering ensures that records cannot be deleted or reports destroyed.  Many organizations prefer not to prosecute employees suspected of fraudulent behavior because of the negative publicity they will receive in the press. This in itself encourages criminals to repeat the activity in their new working environment knowing the likelihood of punishment is remote.
Unauthorized Data Access
Password protection is the most common method of protecting corporate data. Nevertheless, fraudulent transactions are often carried out by unauthorized users who manage to gain access to the corporate network by using the login details of another user. One way of achieving this is through a terminal spoof – a simple yet effective approach to finding other user’s passwords. A terminal spoof is a program that runs on a machine and looks like the normal login screen. Once a user has given his or her user-id and password, the terminal spoof will record both on the local disk or server, give what looks like an authentic error message (such as invalid password – please re-enter) and then passes control to the real login program. The criminal will pick up the passwords later to gain access to the system masquerading as the unfortunate victim.
Other criminals simply make use of an unattended computer that has been left on by a user who has logged in to the network and then left the office. Time-out or screen-saver programs with password protection provide a simple barrier to this approach In addition, locked doors are a traditional means of excluding undesirable visitors.
Other dangers of which managers should be aware include the Trojan horse , in which code is added to a program, which will activate under certain conditions. For example, a computer consultant in Johannesburg had a client in Durban. He placed a Trojan horse in the payroll program so that it would malfunction while processing the June payroll. They would fly him down, all expenses paid, to fix the problem and stay for the Durban July horse race. Once this had happened for the third time, another consultant was used who uncovered the offending code.
Another risk is the Back-door technique . When programmers are building systems, they may try to bypass all the access security procedures to speed up the development time. In some cases, these “back doors” have not been removed and the programmer can gain illegal entry into the production system.
Sabotage and Theft
When computers were the size of small houses and hidden in secure computer installations, then theft of computer hardware was rare. Today, PCs are on most desks and in many cases they have to be physically bolted to the table to prevent their disappearance. One famous case of theft involved a laptop computer stolen from the back seat of a car in the USA in early 1991. On the hard disk was the master plan for Desert Storm, the details of how the United States and her allies would attack Iraq.
Mobile computing devices are especially vulnerable to theft, and limiting of physical access to equipment is the most effective first line of defense. Restrictions to entry can be based on electronic locks, activated by means of magnetic disks or swipe cards, or on advanced biometric devices that identify the individual based on characteristics such as fingerprints or the pattern of the retina. (In each case, the security mechanism would obviously be linked to a database containing details of authorized users.)
Another form of theft relates to the copying of programs and data resources in an organization. Obtaining customer lists together with the details of the amount and type of business can obviously assist companies to encourage customers away from their competition. Theft of software is a major problem in the PC world where users often make illegal copies of the programs rather than purchase the package themselves – this practice is known as software piracy. This type of theft is more difficult to identify, since the original product has not physically disappeared as with the theft of computer hardware. Where software piracy is discovered, the owner of the computer on which the software resides (often the employer) is held to be legally responsible for the presence of pirated software.
The last category of computer theft covers the illegal use of computer time. In the past computer operators were often caught processing work for third parties or users were doing their own work at the office. Computer hackers spend their time searching for networks to which they can gain access. Having breached the security controls, they often browse around the databases in the installation but may not do any damage. In these instances, the only crime they can be charged with is the theft of computer time.
Security Beyond the Organization
Hackers and Firewalls
Hackers are users from outside the organization, who penetrate a computer system usually through its communication lines. Although some hackers are content merely to demonstrate that they have bypassed network security, there is a high risk of malicious damage to data, stealing of sensitive information (such as customers’ credit card numbers) or entry of fraudulent transactions by the hacker. Hackers may also instigate a denial-of-service attack, in which a targeted web site is inundated with requests for information initiated by the hacker, rendering it inaccessible for genuine business customers.
A firewall is an additional system that enforces access control policy between two networks, especially between a corporate network and the Internet. The firewall monitors all external communications, checks user authorization and maintains a log of all attempts to access the network. They can also be used to check for the presence of viruses, for the downloading of unauthorized software, and to guard against denial-of-service attacks.
Data which is in the process of being communicated is also vulnerable to eavesdropping. Encryption, which scrambles data into an unreadable form, can be used to improve data privacy and prevent any unauthorized changes to the message, as well as protecting the confidentiality of data within the organization.
Viruses
A computer virus is a program that invades a computer system, normally by residing in corrupt files. The virus has the ability to replicate itself and so spread to other files and computer systems.
There are three main types of viruses. The original viruses were mainly systems viruses. They resided in the boot sector of a disk (the first place the computer looks when loading the operating system) or in the operating system utilities. These viruses were usually easy to find and clean. One problem was that they loaded into memory as soon as an infected machine was switched on and could often hide themselves from the anti-viral software.
The next generation of viruses attached themselves to executable files. When an infected program is run, the virus resides in memory and infects all new programs run until the system is switched off. These viruses are difficult to counter, especially on a network as common files are often infected. Even when the file server is cleared of infected programs, users have copies on their personal hard drives, or a infected copy of the program in memory when the virus check is performed on the server.
One area most users with which felt safe was the accessing of non-executable files such as word processing documents. Unfortunately there are now a number of macro viruses that can attach themselves to documents and spreadsheets. Even receiving a simple letter as an e-mail attachment can infect your machine. Some of the more well known viruses include:
- Michelangelo. This virus infected many machines in the early months of 1992. The virus was primed to activate on Michelango’s birthday (6th March) and had the capability of destroying all files on the hard disk of infected PC’s. News of the virus made headlines and many businessmen and home users rushed out to purchase programs to check and clean viruses from their installations. On the day, some infections did occur but the main winners were the vendors of virus detection software.
- Stoned. A very common virus in the late 1980s it came in many forms, some harmless while others corrupted files by attacking the directories and allocation tables. The main theme of the virus was to legalize the smoking of cannabis and normally the message “Your PC is now Stoned” would appear on the screen.
- Jerusalem. Deletes all programs that are run on Friday 13th.
- Concept. A macro virus, attached to word processing documents.
- EXEBUG. A nasty little bug that may corrupt your hard drive. This is a systems virus and can infect the CMOS of your PC. Very difficult to find and eradicate as it uses “stealth” technology to hide from virus checkers.
With increasing globalization and interorganizational communications, viruses are able to spread faster and further than ever before. Recent examples include Melissa, ILoveYou, and the Nimda virus, which make use of multiple methods of transmission. The real mystery about viruses is why they exist at all. Some experts suggest that computer hackers are motivated by the challenge of “beating the system” by writing programs that can bypass virus protection systems and take control of each individual machine. A more likely motive is to induce computer users to purchase legal copies of software. The only real winners in the new world of computer viruses are the companies selling computer software.
In the early 1980’s the illegal copying of PC software, or software piracy was rife as there was no real business advantage to purchasing the software (except a copy of the official reference manual). Today it is estimated that about 50% of the PC software used in the USA has been illegally copied and the situation in South Africa is likely to be worse. One of the major motivations for buying software rather than copying from a friend is that the shrink-wrapped product is guaranteed to be virus free.
The best line of defense against viruses is the regular use of up-to-date anti-virus software, which will scan files for viruses and remove them if found.
- Discovering Information Systems Sections 10.1-10.3. Authored by : Jean-Paul Van Belle, Mike Eccles, and Jane Nash. Located at : https://archive.org/details/ost-computer-science-discoveringinformationsystems . License : CC BY-NC-ND: Attribution-NonCommercial-NoDerivatives
- DMZ Network Diagram Two Firewalls. Authored by : Jasonlfunk. Located at : https://en.wikipedia.org/wiki/File:DMZ_network_diagram_2_firewalls.png . License : CC0: No Rights Reserved
- Vulnerable. Authored by : Jannis Blume. Located at : https://www.flickr.com/photos/iblume/6230865185/ . License : CC BY-SA: Attribution-ShareAlike
- I, the Hacker. Authored by : Charis Tsevis. Located at : https://www.flickr.com/photos/tsevis/93150384/ . License : CC BY-NC-ND: Attribution-NonCommercial-NoDerivatives


6.1: Basic Notation and Terminology
A partially ordered set or poset \(P\) is a pair \((X,P)\) where \(X\) is a set and \(P\) is a reflexive, antisymmetric, and transitive binary relation on \(X\). (Refer to Section B.10 for a refresher of what these properties are if you need to.) We call \(X\) the ground set while \(P\) is a partial order on \(X\). Elements of the ground set \(X\) are also called points , and the poset \(P\) is finite if its ground set \(X\) is a finite set.
Let \(X=\{a,b,c,d,e,f\}\). Consider the following binary relations on \(X\).
\(R_1=\{(a,a),(b,b),(c,c),(d,d),(e,e),(f,f),(a,b),(a,c),(e,f)\}\)
\(R_2=\{(a,a),(b,b),(c,c),(d,d),(e,e),(f,f),(d,b),(d,e),(b,a),(e,a),(d,a),(c,f)\}\)
\(R_3=\{(a,a),(b,b),(c,c),(d,d),(e,e),(f,f),(a,c),(a,e),(a,f),(b,c),(b,d),(b,e),(b,f),(d,e),(d,f),(e,f)\}\)
\(R_4=\{(a,a),(b,b),(c,c),(d,d),(e,e),(f,f),(d,b),(b,a),(e,a),(c,f)\}\)
\(R_5=\{(a,a),(c,c),(d,d),(e,e),(a,e),(c,a),(c,e),(d,e)\}\)
\(R_6=\{(a,a),(b,b),(c,c),(d,d),(e,e),(f,f),(d,f),(b,e),(c,a),(e,b)\}\)
Which of the binary relations are partial orders on \(X\)? For those that are not partial orders on \(X\), which property or properties are violated?
- Solution
-
A bit of checking confirms that \(R_1, R_2\) and \(R_3\) are partial orders on \(X\), so \(P_1=(X,R_1), P_2=(X,R_2)\) and \(P_3=(X,R_3)\) are posets. Several of the other examples we will discuss in this chapter will use the poset \(P_3=(X,R_3)\).
On the other hand, \(R_4, R_5\) and \(R_6\) are not partial orders on \(X\). Note that \(R_4\) is not transitive, as it contains (\(d,b\)) and (\(b,a\)) but not \((d,a)\). The relation \(R_5\) is not reflexive, since it doesn't contain \((b,b)\). (Also, it also doesn't contain \((f,f)\), but one shortcoming is enough.) Note that \(R_5\) is a partial order on \(\{a,b,d,e\}\). The relation \(R_6\) is not antisymmetric, as it contains both \((b,e)\) and \((e,b)\).
When \(P=(X,P)\) is a poset, it is common to write \(x \leq y\) in \(P\) or \(y \geq x\) in \(P\) as substitutes for \((x,y) \in P\). Of course, the notations \(x<y\) in \(P\) and \(y>x\) in \(P\) mean \(x \leq y\) in \(P\) and \(x \neq y\). When the poset \(P\) remains fixed throughout a discussion, we will sometimes abbreviate \(x \leq y\) in \(P\) by just writing \(x \leq y\), etc. When \(x\) and \(y\) are distinct points from \(X\), we say \(x\) is covered by \(y\) in \(P\) when \(x<y\) in \(P\), and there is no point \(z \in X\) for which \(x<z\) and \(z<y\) in \(P\). For example, in the poset \(P_3=(X,R_3)\) from Example 6.3 , \(d\) is covered by \(e\) and \(c\) covers \(b\). However, \(a\) is not covered by \(f\), since \(a<e<f\) in \(R_3\). We can then associate with the poset \(P\) a cover graph \(G\) whose vertex set is the ground set \(X\) of \(P\) with \(xy\) an edge in \(G\) if and only if one of \(x\) and \(y\) covers the other in \(P\). Again, for the poset \(P_3\) from Example 6.3 , we show the cover graph on the left side of Figure 6.4 . Actually, on the right side of this figure is just another drawing of this same graph.
It is convenient to illustrate a poset with a suitably drawn diagram of the cover graph in the Euclidean plane. We choose a standard horizontal/vertical coordinate system in the plane and require that the vertical coordinate of the point corresponding to \(y\) be larger than the vertical coordinate of the point corresponding to \(x\) whenever \(y\) covers \(x\) in \(P\). Each edge in the cover graph is represented by a straight line segment which contains no point corresponding to any element in the poset other than those associated with its two end points. Such diagrams are called Hasse diagrams ( poset diagrams, order diagrams , or just diagrams ). Now it should be clear that the drawing on the right side of Figure 6.4 is a diagram of the poset \(P_3\) from Example 6.3 , while the diagram on the left is not.
For posets of moderate size, diagrams are frequently used to define a poset—rather than the explicit binary relation notation illustrated in Example 6.3 . In Figure 6.5 , we illustrate a poset \(P=(X,P)\) with ground set \(X=[17]=\{1,2,…,17\}\). It would take several lines of text to write out the binary relation \(P\), and somehow the diagram serves to give us a more tactile sense of the properties of the poset.
Alice and Bob are talking about how you communicate with a computer in working with posets. Bob says that computers have incredible graphics capabilities these days and that you just give the computer a pdf scan of a diagram. Alice says that she doubts that anybody really does that. Carlos says that there are several effective strategies. One way is to label the points with positive integers from \([n]\) where \(n\) is the number of points in the ground set and then define a 0–1 \(n \times n\) matrix A with entry \(a(i,j)=1\) when \(i \leq j\) in \(P\) and \(a(i,j)=0\) otherwise. Alternatively, you can just provide for each element \(i\) in the ground set a vector \(U(x)\) listing all elements which are greater than \(x\) in \(P\). This vector can be what computer scientists call a linked list .
A partial order \(P\) is called a total order (also, a linear order ) if for all \(x,y \in X\), either \(x \leq y\) in \(P\) or \(y \leq x\) in \(P\). For small finite sets, we can specify a linear order by listing the elements from least to greatest. For example, \(L=[b,c,d,a,f,g,e]\) is the linear order on the ground set \(\{a,b,c,d,e,f,g\}\) with \(b<c<d<a<f<g<e\) in \(L\).
The set of real numbers comes equipped with a natural total order. For example, \(1<7/5<\sqrt{2}< \pi \) in this order. But in this chapter, we will be interested primarily with partial orders that are not linear orders. Also, we note that special care must be taken when discussing partial orders on ground sets whose elements are real numbers. For the poset shown in Figure 6.5 , note that 14 is less than 8, while 3 and 6 are incomparable. Best not to tell your parents that you've learned that under certain circumstances, 14 can be less than 8 and that you may be able to say which of 3 and 6 is larger than the other. The subtlety may be lost in the heated discussion certain to follow.
There are several quite natural ways to construct posets.
- A family \(\mathcal{F}\) of sets is partially ordered by inclusion, i.e., set \(A \leq B\) if and only if \(A\) is a subset of \(B\).
- A set \(X\) of positive integers is partially ordered by division—without remainder, i.e., set \(m \leq n\) if and only if \(n \equiv 0\) (mod \(m\)).
-
A set \(X\) of \(t\)-tuples of real numbers is partially ordered by the rule
\((a_1,a_2,…,a_t) \leq (b_1,b_2,…,b_t)\)
if and only if \(a_i \leq b_i\) in the natural order on \(\mathcal{R}\) for \(i = 1,2,...,t\).
- When \(L_1, L_2,…,L_k\) are linear orders on the same set \(X\), we can define a partial order \(P\) on \(X\) by setting \(x \leq y\) in \(P\) if and only if \(x \leq y\) in \(L_i\) for all \(i=1,2,…,k\).
We illustrate the first three constructions with the posets shown in Figure 6.8 . As is now clear, in the discussion at the very beginning of this chapter, Dave drew a diagram for the poset determined by the intersection of the linear orders given by Alice and the movie critic.
Distinct points \(x\) and \(y\) in a poset \(P=(X,P)\) are comparable if either \(x<y\) in \(P\) or \(x>y\) in \(P\); otherwise \(x\) and \(y\) are incomparable . If \(x\) and \(y\) are incomparable in \(P\), we sometimes write \(x‖y\) in \(P\).With a poset \(P=(X,P)\), we associate a comparability graph \(G_1=(X,E_1)\) and an incomparability graph \(G_2=(X,E_2)\). The edges in the comparability graph \(G_1\) consist of the comparable pairs and the edges in the incomparability graph are the incomparable pairs. We illustrate these definitions in Figure 6.9 where we show the comparability graph and the incomparability graph of the poset \(P_3\).
When \(P=(X,P)\) is a poset and \(Y \subseteq X\), the binary relation \(Q=P \bigcap (Y \times Y)\) is a partial order on \(Y\), and we call the poset \((Y,Q)\) a subposet of \(P\). In Figure 6.10 , we show a subposet of the poset first presented in Figure 6.5 .
When \(P=(X,P)\) is a poset and \(C\) is a subset of \(X\), we say that \(C\) is a chain if every distinct pair of points from \(C\) is comparable in \(P\). When \(P\) is a linear order, the entire ground set \(X\) is a chain. Dually, if \(A\) is a subset of \(X\), we say that \(A\) is an antichain if every distinct pair of points from \(A\) is incomparable in \(P\). Note that a one-element subset is both a chain and an antichain. Also, we consider the empty set as both a chain and an antichain.
The height of a poset \(P=(X,P)\), denoted height(P), is the largest \(h\) for which there exists a chain of \(h\) points in \(P\). Dually, the width of a poset \(P=(X,P)\), denoted width(P), is the largest \(w\) for which there exists an antichain of \(w\) points in \(P\).
Given a poset \(P=(X,P)\), how hard is to determine its height and width? Bob says that it is very easy. For example, to find the width of a poset, just list all the subsets of \(X\). Delete those which are not antichains. The answer is the size of the largest subset that remains. He is quick to assert that the same approach will work to find the height. Alice groans at Bob's naivety and suggests that he should read further in this chapter.