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Questions and Answers of
Statistics
(a) Show that p ( [q( ( p ˄ q)] (b) True or false? p([q ( (p ˄ q)]
1. True or false? (p ˅ q) ( [q ( (p ˄ q)] 2. True or false? [p ˅ ( p ˄ q)] ( p
Write a statement equivalent to ( (p ˅ q) ˄ (q, Using only ( and (?
Show that the argument is valid. 1. The auto manufacturer follows environmental regulations or they face legal action. The manufacturer does not follow environmental regulations. Therefore, the
Test the validity of the arguments.1. If the salaries go up, then more people apply. More people apply or the salaries go up. Therefore, the salaries go up.2. If Rita studies, she gets good grades.
Use indirect proof to show that the argument is valid. 1. Sam goes to the store only if he needs milk. Sam does not need milk. Therefore, Sam does not go to the store. 2. If it rains hard, there will
Show that each of the arguments in Exercises 1 and 2 is valid, first by using a direct proof and then by using an indirect proof.1. If Jimmy does not find the keys, he will do his homework. Jimmy
Show that each of the arguments in Exercises 1 and 2 is valid by using a direct proof or by using an indirect proof.1. If Marissa does not go to the movies, then she is not idle. If Marissa is not in
Let U = {1, 2, 3, 4, 5, 6}. Determine the truth value of p(x) = [(x is even) or (x is divisible by 3)] for the given values of x. (a) x = 1 (b) x = 4 (c) x = 3 (d) x = 6 (e) x = 5
Let the universe consist of all real numbers. Let p(x) = (x is positive) and q(x) = (x is a perfect square). Determine the truth value of each of the following: (a) (x p(x) (b) (x q(x) (c) ∃x p(x)
Negate each statement by changing existential quantifiers to universal quantifiers, or vice versa. (a) Every dog has its day. (b) Some men fight wars. (c) All women are married. (d) For every pot,
Negate each statement by changing existential quantifiers to universal quantifiers, or vice versa. (a) Every stitch saves time. (b) All books have hard covers. (c) Some children are afraid of
Consider the universe of nonnegative integers = {0, 1, 2, 3, 4, . . . .}. Write the English sentence for each symbolic statement. Determine the truth value of each statement. If the statement is
Consider the universe of all subsets of the set A = {a, b, c}. Let the variables x and y denote subsets of A. Find the truth value of each of the statements, and explain your answer. If the statement
Let the universe for both variables x and y be the set {1, 2, 3, 4, 5, 6}. Let p(x, y) = "x divides y." Give the truth values of each of the statements; explain your answer, and give a counterexample
1. If p(x) denotes "x ∈ S" and q(x) denotes "x ∈ T," describe the following, using logical statement forms with p(x) and q(x): (a) x ∈ S' ( T' (b) x ∈ S ( T ′ (c) x ∈ S′ ( T ′ (d) x
1. Is S a subset of T? Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} S = {x ∈ U: x divides 12 evenly} T = {x ∈ U: x is a multiple of 2}. 2. Prove that S is a subset of T, where U = {1, 2, 3, 4,
Determine the truth value of p(x) for the values of x chosen from the universe of all letters of the alphabet, where p(x) = [(x is a vowel) and (x is in the word ABLE)]. (a) x = A (b) x = D (c) x = B
1. An alert California teacher chided "Dear Abby" (Baltimore Sun, March 1, 1989) for her statement "Confidential to Eunice: All men do not cheat on their wives." Let p(x) be the statement "x cheats
Let the universe be all university professors. Let p(x) be the open statement "x likes poetry." Write the following statements symbolically: (a) All university professors like poetry. (b) Some
Consider the universe of all orange juice. Write the following symbolically, using p(x) = "x comes from Florida." (a) All orange juice does not come from Florida. (b) Some orange juice comes from
1. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let p(x) = [(x is prime) ( (x2 + 1 is even)]. Find the truth value of (a) ∃x p(x) (b) (x p(x) 2. Let U = {0, 1, 2, 3, 4}. Let p(x) = [x2 > 9]. Find the truth
Let the universe consist of all nonnegative integers. Let p(x) be the statement "x is even." Let q(x) be the statement "x is odd." Determine the truth value of (a) (x [p(x) ˅ q(x)] (b) [(x p(x)] ˅
1. Write the logic statement represented by Fig. 10.2. Write the logic statement represented by Fig. 11. 3. Write the logic statement represented by Fig. 12. 4. Write the logic statement represented
Simplify the logic circuit in Fig. 15 as much as possible.Figure 15:
Simplify the logic circuit in Fig. 16 as much as possible.Figure 16:
1. Simplify the logic circuit in Fig. 17 as much as possible.Figure 17:2. Design a logic circuit with two inputs (use p and q) that always has output 1?
1. Design a logic circuit with two inputs (use p and q) that always has output 0. 2. Design a logic circuit that acts as an or gate for inputs p and q, using three nor gates?
1. Design a logic circuit that acts as an and gate for inputs p and q, using three NAND gates. 2. Design a logic circuit that acts as an XOR gate for inputs p and q, using three gates other than XOR
Design a logic circuit for (p NAND q) NAND (p NAND q), and show the outputs for all values of p and q on a truth table?
1. In the game of "match," each of two players, A and B, has control of an input switch. At the signal, each player allows the current to flow (on) or stops the current (off). If both players choose
Draw the logic circuit that represents each of the logical statements in Exercises 5-8. Determine the output when the variables have the values given? 1. p ˄ ((q ˅ r); for p = 1, q = 0, r = 1 2. (p
Simplify the logic circuit in Fig. 14 as much as possible.Figure 14:
Carbon monoxide molecules happen to be just the right size and shape, and happen to have just the right chemical properties, to fit neatly into cavities within hemoglobin molecules in blood that are
Punishment, when speedy and specific, may suppress undesirable behavior, but it cannot teach or encourage desirable alternatives. Therefore, it is crucial to use positive techniques to model and
Profit serves a very crucial function in a free-enterprise economy, such as our own. High profits are the signal that consumers want more of the output of the industry. High profits provide the
Cats can think circles around dogs! My cat regularly used to close and lock the door to my neighbor's doghouse, trapping their sleeping Doberman inside. Try telling a cat what to do, or putting a
Since private property helps people define themselves, since it frees people from mundane cares of daily subsistence, and since it is finite, no individual should accumulate so much property that
To every existing thing God wills some good. Hence, since to love any thing is nothing else than to will good to that thing, it is manifest that God loves everything that exists. Each of the
Women of the working class, especially wage workers, should not have more than two children at most. The average working man can support no more and the average working woman can take care of no more
Radioactive fallout isn't the only concern in the aftermath of nuclear explosions. The nations of planet Earth have acquired nuclear weapons with an explosive power equal to more than a million
An ant releases a chemical when it dies, and its fellows then carry it away to the compost heap. Apparently the communication is highly effective; a healthy ant painted with the death chemical will
Every art and every inquiry, and similarly every action and pursuit, is thought to aim at some good; and for this reason the good has rightly been declared to be that at which all things aim. Each of
Poverty offers numerous benefits to the nonpoor. Antipoverty programs provide jobs for middle-class professionals in social work, penology, and public health. Such workers' future advancement is tied
Since the good, according to Plato, is that which furthers a person's real interests, it follows that in any given case when the good is known, men will seek it. Each of the following passages
Corn is an annual crop. Butcher's meat, a crop which requires four or five years to grow. As an acre of land, therefore, will produce a much smaller quantity of the one species of food than the
Neither a borrower nor lender be For loan oft loses both itself and friend, And borrowing dulls the edge of husbandry. Each of the following passages contains a single argument. Using the letters "P"
The stakes in whistle-blowing are high. Take the nurse who alleges that physicians enrich themselves in her hospital through unnecessary surgery; the engineer who discloses safety defects in the
If a piece of information is not "job relevant," then the employer is not entitled qua employer to know it. Consequently, since sexual practices, political beliefs, associational activities, etc.,
Many people believe that a dark tan is attractive and a sign of good health, but mounting evidence indicates that too much sun can lead to health problems. One of the most noticeable effects is
Contrary to the tales of some scuba divers, the toothy, gaping grin on the mouth of an approaching shark is not necessarily anticipatory. It is generally accepted that by constantly swimming with its
Not only is the sky blue [as a result of scattering], but light coming from it is also partially polarized. You can readily observe this by placing a piece of Polaroid (for example, one lens of a
Since the secondary light [from the moon] does not inherently belong to the moon and is not received from any star or from the sun, and since in the whole universe there is no other body left but the
Anyone familiar with our prison system knows that there are some inmates who behave little better than brute beasts. But the very fact that these prisoners exist is a telling argument against the
Though it is possible that REM sleep and dreaming are not necessary in the adult, REM deprivation studies seem to suggest otherwise. Why would REM pressure increase with deprivation if the system is
As the denial or perversion of justice by the sentences of courts, as well as in any other manner, is with reason classed among the just causes of war, it will follow that the federal judiciary ought
We say that an end pursued in its own right is more complete than an end pursued because of something else, and that an end that is never choiceworthy because of something else is more complete than
When individuals voluntarily abandon property, they forfeit any expectation of privacy in it that they might have had. Therefore, a warrantless search or seizure of abandoned property is not
Artists and poets look at the world and seek relationships and order. But they translate their ideas to canvas, or to marble, or into poetic images. Scientists try to find relationships between
The fact that there was never a land bridge between Australia and mainland Asia is evidenced by the fact that the animal species in the two areas are very different. Asian placental mammals and
It really does matter if you get enough sleep. We need sleep to think clearly, react quickly, and create memories. Studies show that people who are taught mentally challenging tasks do better after a
The classroom teacher is crucial to the development and academic success of the average student, and administrators simply are ancillary to this effort. For this reason, classroom teachers ought to
An agreement cannot bind unless both parties to the agreement know what they are doing and freely choose to do it. This implies that the seller who intends to enter a contract with a customer has a
1. Because triangle A is congruent with triangle B, and triangle A is isosceles, it follows that triangle B is isosceles.2. The plaque on the Leaning Tower of Pisa says that Galileo performed
1. Crater Lake, the deepest lake in the United States, was caused by a huge volcanic eruption 7,700 years ago. Since human beings have lived around the mountain for more than 10,000 years, it is
That [the moons of Jupiter] revolve in unequal circles is manifestly deduced from the fact that at the longest elongation from Jupiter it is never possible to see two of these moons in conjunction,
1. Lenses function by refracting light at their surfaces. Consequently, their action depends not only on the shape of the lens surfaces, but also on the indices of refraction of the lens material and
1. The Matterhorn is higher than Mount Whitney, and Mount Whitney is higher than Mount Rainier. The obvious conclusion is that the Matterhorn is higher than Mount Rainier. 2. Although both front and
Define the following terms: Deductive argument..........................Prediction Inductive argument...........................Argument from analogy Argument based on
Answer "true" or "false" to the following statements: 1. In an inductive argument, it is intended that the conclusion contain more information than the premises. 2. In a deductive argument, the
1. Since Moby Dick was written by Shakespeare, and Moby Dick is a sciencefiction novel, it follows that Shakespeare wrote a science-fiction novel.2. Since London is north of Paris and south of
1. Chicago is a city in Michigan and Michigan is part of the United States. Therefore, Chicago is a city in the United States. 2. If Queen Elizabeth was born in Los Angeles, then she is a California
1. The grave marker at Arlington National Cemetery says that John F. Kennedy is buried there. It must be the case that Kennedy really is buried in that cemetery.2. The ebb and flow of the tides has
1. Since Tom is the brother of Agatha, and Agatha is the mother of Raquel, it follows that Tom is the uncle of Raquel.2. When a cook cannot recall the ingredients in a recipe, it is appropriate that
Define the following terms: Valid deductive argument.....................Strong inductive argument Invalid deductive argument...................Weak inductive argument Sound
Answer "true" or "false" to the following statements: 1. Some arguments, while not completely valid, are almost valid. 2. Inductive arguments allow for varying degrees of strength and weakness. 3.
1. All galaxies are structures that contain black holes in the center, so all galaxies are quasars, since all quasars are structures that contain black holes in the center.2. Some evolutionists are
1. All super PACs are unlimited spenders. For this reason, some big-time power brokers are not super PACs, inasmuch as some unlimited spenders are big-time power brokers. 2. No movie producers are
1. Some improvers of humankind are not exploiters of personal information. As a result, some corporate social networks are not improvers of humankind, seeing that all corporate social networks are
1. No patents are measures that discourage research and development, and all patents are regulations that protect intellectual property. Thus, no measures that discourage research and development are
1. Some politicians are people who will stop at nothing to win an election, and no people who will stop at nothing to win an election are true statesmen. Hence, no politicians are true statesmen. 2.
1. No patrons of fast-food restaurants are health-food addicts. Consequently, no patrons of fast-food restaurants are connoisseurs of fine desserts, since no connoisseurs of fine desserts are
1. All persons who assist others in suicide are people guilty of murder. Accordingly, some individuals motivated by compassion are not persons guilty of murder, inasmuch as some people who assist
1. If animal species are fixed and immutable, then evolution is a myth. Therefore, evolution is not a myth, since animal species are not fixed and immutable. 2. If carbon dioxide is present in the
1. If human rights are recognized, then civilization flourishes. If equality prevails, then civilization flourishes. Thus, if human rights are recognized, then equality prevails.2. If energy taxes
1. All homeless people who are panhandlers are destitute individuals. Therefore, all homeless people are destitute individuals.2. Some wrestlers are colorful hulks, since some wrestlers are colorful
1. All community colleges with low tuition are either schools with large enrollments or institutions supported by taxes. Therefore, all community colleges are institutions supported by taxes.2. All
1. All diabetes victims are either insulin takers or glucose eliminators. Accordingly, some diabetes victims are glucose eliminators, since some diabetes victims are insulin takers.2. All FHA loans
The following arguments were taken from magazine and newspaper editorials and letters to the editor. In most instances the main conclusion must be rephrased to capture the full intent of the author.
1. (1) The conditions under which many food animals are raised are unhealthy for humans. (2) To keep these animals alive, large quantities of drugs must be administered. (3) These drugs remain in the
1. (1) There is a lot of pressure on untenured college teachers to dumb down their courses. (2) Administrators tend to rehire teachers who bring in more money, and (3) teachers who dumb down their
1. (1) Numerous cancer patients on chemotherapy suffer from constant nausea, loss of appetite, and inability to sleep. (2) For many of these patients, inhaled marijuana is the only medicine that
1. (1) When parents become old and destitute, the obligation of caring for them should be imposed on their children. (2) Clearly, children owe a debt to their parents. (3) Their parents brought them
1. (1) It appears that animals may be able to predict earthquakes. (2) Prior to a major quake in China, hundreds of snakes suddenly appeared from hibernation and froze to death in the snow, (3) fish
1. (1) Many people believe that the crime of bribery cannot extend to campaign contributions. (2) From a legal standpoint, however, countless campaign contributions are in fact bribes. (3) A bribe is
1. (1) It has been widely acknowledged that the quality of undergraduate education in this country is diminishing. (2) An often unrecognized cause of this malady is the exploitative way that
1. (1) The state has a right to intervene on behalf of unborn children, and (2) this right should be implemented immediately. (3) While it may be true that a mere fetus has no rights, (4) surely a
1. (1) Powerful new particle accelerators are important in high-energy physics, and (2) they are worth their cost because (3) they will allow scientists to produce and capture significant quantities
1. (1) Cigarette consumption could be easily reduced by simply outlawing tailormade cigarettes. (2) The manufacture of tailor-made cigarettes to American standards is a high-tech industry. (3) It
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