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- Question : 1E - a) Define the negation of a proposition. b) What is the negation of
- Question : 2E - a) Define (using truth tables) the disjunction, conjunction, exclusive or, conditional, and biconditional of the propositions p and q. b) What are the disjunction, conjunction, exclusive or, conditional, and biconditional of the propositions
- Question : 3E - a) Describe at least five different ways to write the conditional statement p ? q in English. b) Define the converse and contrapositive of a conditional statement. c) State the converse and the contrapositive of the conditional statement
- Question : 4E - a) What does it mean for two propositions to be logically equivalent? b) Describe the different ways to show that two compound propositions are logically equivalent. c) Show in at least two different ways that the compound propositions
- Question : 5E - (Depends on the Exercise Set in Section 1.3) a) Given a truth table, explain how to use disjunctive normal form to construct a compound proposition with this truth table. b) Explain why part (a) shows that the operators ?, ?, and
- Question : 6E - What are the universal and existential quantifications of a predicate P (x)? What are their negations?
- Question : 7E - a) What is the difference between the quantification ?x?yP (x, y) and ?y?xP (x, y), where P (x, y) is a predicate? b) Give an example of a predicate P (x, y) such that ?x?yP (x, y) and ?y?xP (x, y) have different truth values.
- Question : 8E - Describe what is meant by a valid argument in propositional logic and show that the argument
- Question : 9E - Use rules of inference to show that if the premises
- Question : 10E - a) Describe what is meant by a direct proof, a proof by contraposition, and a proof by contradiction of a conditional statement p ? q. b) Give a direct proof, a proof by contraposition and a proof by contradiction of the statement:
- Question : 11E - a) Describe a way to prove the biconditional p ? q. b) Prove the statement:
- Question : 12E - To prove that the statements p1, p2, p3, and p4 are equivalent, is it sufficient to show that the conditional statements p4 ? p2, p3 ? p1, and p1 ? p2 are valid? If not, provide another collection of conditional statements that can be used to show that the four statements are equivalent.
- Question : 13E - a) Suppose that a statement of the form ?xP (x) is false. How can this be proved? b) Show that the statement
- Question : 14E - What is the difference between a constructive and nonconstructive existence proof? Give an example of each.
- Question : 15E - What are the elements of a proof that there is a unique element x such that P (x), where P (x) is a propositional function?
- Question : 16E - Explain how a proof by cases can be used to prove a result about absolute values, such as the fact that |xy|= |x||y| for all real numbers x and y.
- Question : 17E - Let p be the proposition
- Question : 18E - Find the truth table of the compound proposition (p ? q) ? (p ?
- Question : 19E - Show that these compound propositions are tautologies. a) (
- Question : 20E - Give the converse, the contrapositive, and the inverse of these conditional statements. a) If it rains today, then I will drive to work. b) If |x|= x, then x ? 0. c) If n is greater than 3, then n2 is greater than 9.
- Question : 21E - Given a conditional statement p ? q, find the converse of its inverse, the converse of its converse, and the converse of its contrapositive.
- Question : 22E - Given a conditional statement p ? q, find the inverse of its inverse, the inverse of its converse, and the inverse of its contrapositive.
- Question : 23E - Find a compound proposition involving the propositional variables p, q, r, and s that is true when exactly three of these propositional variables are true and is false otherwise.
- Question : 24E - Show that these statements are inconsistent:
- Question : 25E - Show that these statements are inconsistent:
- Question : 26E - Suppose that you meet three people Aaron, Bohan, and Crystal. Can you determine what Aaron, Bohan, and Crystal are if Aaron says
- Question : 27E - Suppose that you meet three people, Anita, Boris, and Carmen. What are Anita, Boris, and Carmen if Anita says
- Question : 28E - (Adapted from [Sm78]) Suppose that on an island there are three types of people, knights, knaves, and normals (also known as spies). Knights always tell the truth, knaves always lie, and normals sometimes lie and sometimes tell the truth. Detectives questioned three inhabitants of the island
- Question : 29E - Show that if S is a proposition, where S is the conditional statement
- Question : 30E - Show that the argument with premises
- Question : 31E - Suppose that the truth value of the proposition pi is T whenever i is an odd positive integer and is F whenever i is an even positive integer. Find the truth values propositional logic ability of a student via a technique known of V100 100 as an obligato game. In an obligato game, a number of rounds is set and in each round the teacher gives the student successive assertions that the student must either accept or reject as they are given. When the student accepts an assertion, it is added as a commitment; when the student rejects an assertion its negation is added as a commitment. The student passes the test if the consistency of all commitments is maintained throughout the test.
- Question : 32E - Suppose that in a three-round obligato game, the teacher first gives the student the proposition p ? q, then the proposition
- Question : 33E - Suppose that in a four-round obligato game, the teacher first gives the student the proposition
- Question : 34E - Explain why every obligato game has a winning strategy. Exercises 13 and 14 are set on the island of knights and knaves described in Example 7 in Section 1.2. i=1(pi ? pi+1) and /i=1(pi ? pi+1).
- Question : 35E - Model 16
- Question : 36E - Let P (x) be the statement
- Question : 37E - Let P (m, n) be the statement
- Question : 38E - Find a domain for the quantifiers in ?x?y(x /= y ? ?z((z = x) ? (z = y))) such that this statement is true.
- Question : 39E - Find a domain for the quantifiers in ?x?y(x /= y ? ?z((z = x) ? (z = y))) such that this statement is false.
- Question : 40E - Use existential and universal quantifiers to express the statement
- Question : 41E - Use existential and universal quantifiers to express the statement
- Question : 42E - The quantifier ?n denotes
- Question : 43E - Express each of these statements using existential and universal quantifiers and propositional logic where ?n is defined in Exercise 26.
- Question : 44E - Express this statement using quantifiers:
- Question : 45E - Express this statement using quantifiers:
- Question : 46E - Express the statement
- Question : 47E - Describe a rule of inference that can be used to prove that there are exactly two elements x and y in a domain such that P (x) and P (y) are true. Express this rule of inference as a statement in English.
- Question : 48E - Use rules of inference to show that if the premises ?x(P (x) ? Q(x)), ?x(Q(x) ? R(x)), and
- Question : 49E - Prove that if x3 is irrational, then x is irrational. ? a) ?0xP (x) b) ?1xP (x) c) ?2xP (x) d) ?3xP (x)
- Question : 50E - Prove that if x is irrational and x ? 0, then tional. x is irra-Let P (x, y) be a propositional function. Show that
- Question : 51E - Prove that given a nonnegative integer n, there is a unique ?x ?y P (x, y) ? ?y ?x P (x, y) is a tautology. nonnegative integer m such that m2 ? n < (m + 1) . Let P (x) and Q(x) be propositional functions. Show that ?x (P (x) ? Q(x)) and ?x P (x) ? ?x Q(x) always
- Question : 52E - Prove that there exists an integer m such that m2 Is your proof constructive or nonconstructive? > 10 1000. have the same truth value.
- Question : 53E - If ?y ?x P (x, y) is true, does it necessarily follow that ?x ?y P (x, y) is true?
- Question : 54E - If ?x ?y P (x, y) is true, does it necessarily follow that ?x ?y P (x, y) is true?
- Question : 55E - Find the negations of these statements. a) If it snows today, then I will go skiing tomorrow. b) Every person in this class understands mathematical induction. c) Some students in this class do not like discrete mathematics. d) In every mathematics class there is some student who falls asleep during lectures.
- Question : 56E - Prove that there is a positive integer that can be written as the sum of squares of positive integers in two different ways. (Use a computer or calculator to speed up your work.)
- Question : 57E - Disprove the statement that every positive integer is the sum of the cubes of eight nonnegative integers.
- Question : 58E - Disprove the statement that every positive integer is the sum of at most two squares and a cube of nonnegative integers.
- Question : 59E - Disprove the statement that every positive integer is the sum of 36 fifth powers of nonnegative integers.
- Question : 60E - Assuming the truth of the theorem that states that ?n is irrational whenever n is a positive integer that is not a perfect square, prove that ?2 + ?3 is irrational.

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