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Discrete and Combinatorial Mathematics An Applied Introduction 5th edition Ralph P. Grimaldi - Solutions

Dustin has a set of 180 distinct blocks. Each of these blocks is made of either wood or plastic and comes in one of three sizes (small, medium, large), five colors (red, white, blue, yellow, green), and six shapes (triangular, square, rectangular, hexagonal, octagonal, circular). How many of the
Verify that [p → (q → r)] → [(p → q) → (p → r)] is a tautology.
Determine all truth value assignments, if any, for the primitive statements p, q, r, s, t that make each of the following compound statements false. (a) [(p ∧ q)∧ r] → (s ∨ t)
The integer variables m and n are assigned the values 3 and 8, respectively, during the execution of a program (written in pseudocode). Each of the following successive statements is then encountered during program execution. [Here the values of m, n following the execution of the statement in part
In the following program segment i, j, m, and n are integer variables. The values of m and n are supplied by the user earlier in the execution of the total program. for i : = 1 to m do for j : = 1 to n do if i ≠ j then print i + j How many times is the print statement in the segment executed
After baking a pie for the two nieces and two nephews who are visiting her, Aunt Nellie leaves the pie on her kitchen table to cool. Then she drives to the mall to close her boutique for the day. Upon her return she finds that someone has eaten one-quarter of the pie. Since no one was in her house
Let p, q be primitive statements for which the implication p → q is false. Determine the truth values for each of the following. (a) p ∧ q (b) ¬ p ∨ q (c) q → p (d) ¬ q → ¬ p
Let p, q, r denote the following statements about a particular triangle ABC. p: Triangle ABC is isosceles. q: Triangle ABC is equilateral. r: Triangle ABC is equiangular. Translate each of the following into an English sentence, (a) q → p (b) ¬ p → ¬ q (c) q ↔ r (d) p ∧ ¬ q (e) r → p
Rewrite each of the following statements as an implication in the if-then form. (a) Practicing her serve daily is a sufficient condition for Darci to have a good chance of winning the tennis tournament. (b) Mary will be allowed on Larry's motorcycle only if she wears her helmet.
Construct a truth table for each of the following compound statements, where p, q, r denote primitive statements. (a) ¬(p ∨ ¬q) → ¬p (b) p → (q → r) (c) (p → q) → r (d) (p → q) → (q → p) (e) [p ∧ (¬p → q)] → q (f) (p ∧ q) → p (g) q ↔ (¬ p ∨ ¬ q) (h) [(p → q)
Let p, q, r denote primitive statements. (a) Use truth tables to verify the following logical equivalences. (i) p → (q ∧ r) ⇔ (p → q) ∧ (p → r) (ii) [(p ∨ q) → r] ⇔ [(p → r) ∧ (q → r)] (iii) [p → (q ∨ r)] ⇔ [¬r → (p → q)] (b) Use the substitution rules to show
Show that for primitive statements p, q, p ∨ q ⇔ [(p ∧ ¬ q)] ∨ (¬p ∧ q)] ⇔ ¬ (p ↔ q).
Verify that [(p ↔ q) ∧ (q ↔ r) ∧ (r ↔ p)] ⇔ [(p → q) ∧ (q → r) ∧ (r → p)], for primitive statements p, q, and r.
For primitive statements p, q, (a) Verify that p → [q → (p ∧ q)] is a tautology. (b) Verify that (p ∨ q) → [q → q] is a tautology by using the result from part (a) along with the substitution rules and the laws of logic. (c) Is (p ∨ q) → [q → (p ∧ q)] a tautology?
Define the connective "Nand" or "Not . . . and ..." by (p ↑ q) ⇔ ¬(p ∧ q), for any statements p, q. Represent the following using only this connective. (a) ¬ p (b) p ∨ q (c) p ∧ q (d) p → q (e) p ↔ q
The connective "Nor" or "Not ... or ..." is defined for any statements p, q by (p ↑ q) ⇔ ¬ (p ∨ q). Represent the statements in parts (a) through (e) of Exercise 15, using only this connective.
For any statements p, q, prove that (a) ¬ (p ↑ q) ⇔ (¬ p ↑ ¬ q) (b) ¬(p ↑ q) ⇔ (¬ p ↑ ¬ q)
Give the reasons for each step in the following simplifications of compound statements. (a) [(p ∨ q) ∧ (p ∨ ¬ q)] ∨ q Reasons ⇔ [p ∨ (q ∧ ¬ q)] ∨ q ⇔ (p ∨ F0) ∨ q ⇔ p ∨ q (b) [(p ↔ q) ∧ (¬q ∧ (r ∨ ¬ q)] Reasons ⇔ (p → q) ∧ ¬q ⇔ (¬p ∨ q) ∧ ¬q ⇔
Provide the steps and reasons, as in Exercise 18, to establish the following logical equivalences. (a) p ∨ [p ∧ (p ∨ q)] ⇔ p (b) p ∨ q ∨ (¬p ∧ ¬q ∧ r) ⇔ p ∨ q ∨ r (c) [(¬p ∨ ¬q) → (p ∧ q ∧ r)] ⇔ p ∧ q
Verify the first Absorption Law by means of a truth table.
Simplify each of the networks shown in Fig. 2.3.
Use the substitution rules to verify that each of the following is a tautology. (Here p, q, and r are primitive statements.) (a) [p ∨ (q ∧ r)] ∨ ¬[p ∨ (q ∧ r)] (b) [(p ∨ q) → r] ↔ [¬r → ¬(p ∨ q)]
For primitive statements p, q, r, and s, simplify the compound statement [[[(p ∧ q) ∧ r] ∨ [(p ∧ q) ∧ ¬r]] ∨ ¬q] → s.
Negate and express each of the following statements in smooth English. (a) Kelsey will get a good education if she puts her studies before her interest in cheerleading. (b) Norma is doing her homework, and Karen is practicing her piano lessons. (c) If Harold passes his C++ course and finishes his
Negate each of the following and simplify the resulting statement. (a) p ∧ (q ∨ r) ∧ (¬ p ∨ ¬ q ∨ r) (b) (p ∧ q) → r (c) p → (¬q ∧ r) (d) p ∨ q ∨ (¬p ∧ ¬q ∧ r)
(a) If p, q are primitive statements, prove that (¬p ∨ q) ∧ (p ∧ (p ∧ q)) ⇔ (p ∧ q). (b) Write the dual of the logical equivalence in part (a).
Write the dual for (a) q → p, (b) p → (q ∧ r), (c) p ↔ q, and (d) p ∨ q, where p, q, and r are primitive statements.
Write the converse, inverse, and contrapositive of each of the following implications. For each implication, determine its truth value as well as the truth values of its corresponding converse, inverse, and contrapositive. (a) If 0 + 0 = 0, then 1 + 1 = 1. (b) If - 1 < 3 and 3 + 7 = 10, then sin
The following are three valid arguments. Establish the validity of each by means of a truth table. In each case, determine which rows of the table are crucial for assessing the validity of the argument and which rows can be ignored. (a) [p ∧ (p → q) ∧ r] → [(p ∨ q) → r] (b) [[(p ∧ q)
Establish the validity of the following arguments.(a) [(p ˆ§ ¬q) ˆ§ r] †’ [(p ˆ§ r) ˆ¨ q](b) [p ˆ§ (p †’ q) ˆ§ (¬q ˆ¨ r)] †’ r(c)(d) (e) (f) (g) (h)
Show that each of the following arguments is invalid by providing a counterexample - that is, an assignment of truth values for the given primitive statements p, q, r, and 5 such that all premises are true (have the truth value 1) while the conclusion is false (has the truth value 0).(a) [(p
Write each of the following arguments in symbolic form. Then establish the validity of the argument or give a counterexample to show that it is invalid.(a) If Rochelle gets the supervisor's position and works hard, then she'll get a raise. If she gets the raise, then she'll buy a new car. She has
(a) Given primitive statements p, q, r, show that the implication[(p ˆ¨ q) ˆ§ (¬p ˆ¨ r)] †’ (q ˆ¨ r)is a tautology.(b) The tautology in part (a) provides the rule of inference known as resolution, where the conclusion (q ˆ¨ r) is called the resolvent. This rule was proposed in 1965
Use truth tables to verify that each of the following is a logical implication. (a) [(p → q) ∧ (q → r)] → (p → r) (b) [(p → q) ∧ ¬ q] → ¬p (c) [(p ∨ q) ∧ → ¬q] → ¬ q (d) [(p → r) ∧ (q → r)] → [(p ∨ q) → r]
Verify that each of the following is a logical implication by showing that it is impossible for the conclusion to have the truth value 0 while the hypothesis has the truth value 1. (a) (p ∧ q) → p (b) p (p ∨ q) (c) [(p ∨ q)∧ ¬ p] → q (d) [(p → q) ∧ (r → s) ∧ (p ∨ r)] → (q
For each of the following pairs of statements, use Modus Ponens or Modus Tollens to fill in the blank line so that a valid argument is presented. (a) If Janice has trouble starting her car, then her daughter Angela will check Janice's spark plugs. ∴ Janice had trouble starting her car. (b) If
Consider each of the following arguments. If the argument is valid, identify the rule of inference that establishes its validity. If not, indicate whether the error is due to an attempt to argue by the converse or by the inverse. (a) Andrea can program in C++, and she can program in Java. Therefore
For primitive statements p, q, and r, let P denote the statement [p ∧ (q ∧ r)] ∨ ¬[p ∨ (q ∧ r)], while P1 denotes the statement [P ∧ (q ∨ r)] ∨ ¬[p ∨ (q ∨ r)]. (a) Use the rules of inference to show that q ∧ r ⇒ q ∨ r. (b) Is it true that P ⇒ P1?
Give the reason(s) for each step needed to show that the following argument is valid. [p ∧ (p → q) ∧ (s ∨ r) ∧ (r → ¬q)] → (s ∨ t) Steps Reasons (1) p (2) p → q (3) q (4) r → ¬q (5) q → ¬r (6) ¬r (7) s ∨ r (8) s (9) ∴ s ∨ t
Give the reasons for the steps verifying the following argument.Steps Reasons (1) ¬s ˆ§ ¬u (2) ¬u (3) ¬u †’ ¬t (4) ¬t (5) ¬ s (6) ¬s ˆ§ ¬ t (7) r †’ (s ˆ¨ t) (8) ¬(s ˆ¨ t) †’ ¬r (9)
(a) Give the reasons for the steps given to validate the argument [(p → q) ∧ (¬r ∨ s) ∧ (p ∨ r)] → (¬q → s). Steps Reasons (1) ¬ (¬q → s) (2) ¬q ∧ ¬s (3) ¬s (4) ¬r ∨ s (5) ¬ r (6) p → q (7) ¬ q (8) ¬ p (9) p ∨ r (10) r (11) ¬ r ∧ r (12) ∴ ¬q → s (b) Give
Let p(x), q(x) denote the following open statements. p(x): x ≤ 3 q(x): x + 1 is odd If the universe consists of all integers, what are the truth values of the following statements? (a) q(1) (b) ¬ p(3) (c) p(7) ∨ q(7) (d) p(3) ∧ q(4) (e) ¬ (p(- 4) ∨ q(- 3)) (f) ¬ p(- 4) ∧ ¬q (- 3)
For the following program segment, m and n are integer variables. The variable A is a two-dimensional array A[1, 1], A[1, 2], . . . , A[1, 20], . . ., A[10, 1], . . . , A[10, 20], with 10 rows (indexed from 1 to 10) and 20 columns (indexed from 1 to 20). for m : = 1 to 10 do for n : = 1 to 20 do A
(a) Let p(x, y) denote the open statement "x divides y," where the universe for each of the variables x, y comprises all integers. (In this context "divides" means "exactly divides" or "divides evenly.") Determine the truth value of each of the following statements; if a quantified statement is
Suppose that p(x, y) is ∧n open statement where the universe for each of x, y consists of only three integers: 2, 3, 5. Then the quantified statement ∃y p(2, y) is logically equivalent to p(2, 2) ∨ p(2, 3) ∨ p(2, 5). The quantified statement ∃x ∀y p(x, y) is logically equivalent to
Let p(n), q(n) represent the open statements p(n): n is odd q(n): n2 is odd for the universe of all integers. Which of the following statements are logically equivalent to each other? (a) If the square of an integer is odd, then the integer is odd. (b) ∀n [p(n) is necessary for q(n] (c) The
For each of the following pairs of statements determine whether the proposed negation is correct. If correct, determine which is true: the original statement or the proposed negation. If the proposed negation is wrong, write a correct version of the negation and then determine whether the original
Write the negation of each of the following statements as an English sentence - without symbolic notation. (Here the universe consists of all the students at the university where Professor Lenhart teaches.) (a) Every student in Professor Lenhart's C++ class is majoring in computer science or
Write the negation of each of the following true statements. For parts (a) and (b) the universe consists of all integers; for parts (c) and (d) the universe comprises all real numbers. (a) For all integers n, if n is not (exactly) divisible by 2, then n is odd. (b) If k, m, n are any integers where
Negate and simplify each of the following. (a) ∃x [p(x) ∨ q(x)] (b) ∀x [p(x) ∧ ¬q(x)] (c) ∀x [p(x) → q(x)] (d) ∃x [(p(x) ∨ q(x)) → r(x)]
For each of the following statements state the converse, inverse, and contrapositive. Also determine the truth value for each given statement, as well as the truth values for its converse, inverse, and contrapositive. (Here "divides" means "exactly divides.") (a) [The universe comprises all
Let p(x), q(x) be defined as in Exercise 1. Let r(x) be the open statement "x > 0." Once again the universe comprises all integers. (a) Determine the truth values of the following statements. (i) p(3) ∨ [q(3) ∨ ¬r(3)] (ii) p(2) → [q(2) → r(2)] (iii) [p(2) ∧ q (2)] → r(2) (iv) p(0) →
Rewrite each of the following statements in the if-then form. Then write the converse, inverse, and contrapositive of your implication. For each result in parts (a) and (c) give the truth value for the implication and the truth values for its converse, inverse, and contrapositive. [In part (a)
For the following statements the universe comprises all nonzero integers. Determine the truth value of each statement. (a) ∃x ∃y [xy = 1] (b) ∃x ∀y [xy = 1] (c) ∀x ∃y [xy = 1] (d) ∃x ∃y [(2x + y = 5) ∧ (x - 3y = - 8)] (e) ∃x ∃y [(3x - y = 1) ∧ (2x + 4y = 3)]
Answer Exercise 21 for the universe of all nonzero real numbers. (a) ∃x ∃y [xy = 1] (b) ∃x ∀y [xy = 1] (c) ∀x ∃y [xy = 1] (d) ∃x ∃y [(2x + y = 5) ∧ (x - 3y = - 8)] (e) ∃x ∃y [(3x - y = 1) ∧ (2x + 4y = 3)]
In the arithmetic of real numbers, there is a real number, namely 0, called the identity of addition because a + 0 = 0 + a = a for every real number a. This may be expressed in symbolic form by ∃z ∀a [a + z = z + a = a]. (We agree that the universe comprises all real numbers.) (a) In
Let the universe for the variables in the following statements consist of all real numbers. In each case negate and simplify the given statement. (a) ∀x ∀y [(x > y) → (x - y > 0)] (b) ∀x ∀y [(x < y) → ∃z (x < z < y)] (c) ∀x ∀y [(|x| = |y|) → (y = ± x)]
Consider the universe of all polygons with three or four sides, and define the following open statements for this universe. a(x): all interior angles of x are equal e(x): x is an equilateral triangle h(x): all sides of x are equal i (x): x is an isosceles triangle p(x): x has an interior angle that
Professor Carlson's class in mechanics is comprised of 29 students of which exactly (1) Three physics majors are juniors; (2) Two electrical engineering majors are juniors; (3) Four mathematics majors are juniors; (4) Twelve physics majors are seniors; (5) Four electrical engineering majors are
Let p(x, y),q(x, y) denote the following open statements. p(x, y): x2 ≥ y q(x, y): x + 2 < y If the universe for each of x, y consists of all real numbers, determine the truth value for each of the following statements. (a) p(2,4) (b) q(1,π) (c) p(- 3, 8) ∧ q(1, 3) (d) p (1/2, 1/3) ∨ ¬q(-
For the universe of all integers, let p(x), q(x), r(x), s(x), and r(x) be the following open statements. p(x): x > 0 q(x): x is even r (x): x is a perfect square s(x): x is (exactly) divisible by 4 f (x) : x is (exactly) divisible by 5 (a) Write the following statements in symbolic form. (i) At
Let p(x), q(x), and r(x) denote the following open statements. p(x): x2 - 8x + 15 = 0 q(x): x is odd r(x): x > 0 For the universe of all integers, determine the truth or falsity of each of the following statements. If a statement is false, give a counterexample. (a) ∀x [p(x) → q(x)] (b) ∀x
Let p(x), q(x), and r(x) be the following open statements. p(x): x2 - 7x + 10 = 0 q(x): x2 - 2x - 3 = 0 r(x): x < 0 (a) Determine the truth or falsity of the following statements, where the universe is all integers. If a statement is false, provide a counterexample or explanation. (i) ∀x [p(x)
In Example 2.52 why did we stop at 26 and not at 28?Suppose that we start with the universe that comprises only the 13 integers 2, 4, 6, 8,..., 24, 26. Then we can establish the statement:For all n (meaning n = 2, 4, 6, ... , 26),we can write n as the sum of at most three perfect squares.The
Provide the missing reasons for the steps verifying the following argument:Steps Reasons (1) ˆ€x [p(x) ˆ¨ q(x)] Premise (2) ˆƒx ¬p(x) Premise (3) ¬ p(a) Step (2) and the definition of the truth for ˆƒx ¬ p(x). [Here a is an element
Write the following argument in symbolic form. Then either verify the validity of the argument or explain why it is invalid. [Assume here that the universe comprises all adults (18 or over) who are presently residing in the city of Las Cruces (in New Mexico). Two of these individuals are Roxe and
Give a direct proof (as in Theorem 2.3) for each of the following. (a) For all integers k and l, if k, l are both even, then k + l is even. (b) For all integers k and l, if k, l are both even, then kl is even.
For each of the following statements provide an indirect proof [as in part (2) of Theorem 2.4] by stating and proving the contrapositive of the given statement. (a) For all integers k and l, if kl is odd, then k, l are both odd. (b) For all integers k and l, if k + l is even, then k and l are both
Prove that for every integer n, if n is odd, then n2 is odd.
Provide a proof by contradiction for the following: For every integer n, if n2 is odd, then n is odd.
Prove that for every integer n, n2 is even if and only if n is even.
Prove the following result in three ways (as in Theorem 2.4): If n is an odd integer, then n + 11 is even.
Let m, n be two positive integers. Prove that if m, n are perfect squares, then the product mn is also a perfect square.
Prove or disprove: If m, n are positive integers and m, n are perfect squares, then m + n is a perfect square.
In Example 2.52 why didn't we include the odd integers between 2 and 26?Suppose that we start with the universe that comprises only the 13 integers 2, 4, 6, 8,..., 24, 26. Then we can establish the statement:For all n (meaning n = 2, 4, 6, ... , 26),we can write n as the sum of at most three
Prove that for all real numbers x and y, if x + y ≥ 100, then x ≥ 50 or y ≥ 50.
Let n be an integer. Prove that n is odd if and only if 7n + 8 is odd.
Let n be an integer. Prove that n is even if and only if 31n + 12 is even.
Use the method of exhaustion to show that every even integer between 30 and 58 (including 30 and 58) can be written as a sum of at most three perfect squares.
Let n be a positive integer greater than 1. We call n prime if the only positive integers that (exactly) divide n are 1 and n itself. For example, the first seven primes are 2, 3, 5, 7, 11, 13, and 17. (We shall learn more about primes in Chapter 4.) Use the method of exhaustion to show that every
For each of the following (universes and) pairs of statements, use the Rule of Universal Specification, in conjunction with Modus Ponens and Modus Tollens, in order to fill in the blank line so that a valid argument results. (a) [The universe comprises all real numbers.] All integers are rational
Determine which of the following arguments are valid and which are invalid. Provide an explanation for each answer. (Let the universe consist of all people presently residing in the United States.) (a) All mail carriers carry a can of mace. Mrs. Bacon is a mail carrier. Therefore Mrs. Bacon carries
For a prescribed universe and any open statements p(x), q(x) in the variable x, prove that (a) ∃x [p(x) ∨ q(x)] ⇔ ∃x [p(x) ∨ ∃x q(x) (b) ∀x [p(x) ∧ q(x) ⇔ ∀x [p(x) ∧ ∀x q(x)
(a) Let p (x), q (x) be open statements in the variable x, with a given universe. Prove that ∀x p(x) ∨ ∀x q(x) ⇒ ∀x [p(x) ∨ q(x)]. [That is, prove that when the statement ∀x p(x) ∨ ∀x q(x) is true, then the statement ∀x [p(x) ∨ g(x)] is true.] (b) Find a counterexample for the
Provide the reasons for the steps verifying the following argument. (Here a denotes a specific but arbitrarily chosen element from the given universe.)Steps Reasons (1) ˆ€x [p(x) †’ (q(x) ˆ§ r(x))] (2) ˆ€x [p(x) ˆ§ s(x)] (3) p(a) †’
Construct the truth table for p ↔ [(q ∧ r) → ¬(s ∨ r)].
Establish the validity of the argument [(p → q) ∧ [(q ∧ r) → s] ∧ r] → (p → s).
Prove or disprove each of the following, where p, q, and r are any statements. (a) [(p ∨ q) ∨ r] ⇔ [p ∨ (q ∨ r)]. (b) [(p ∨(q → r] ⇔ [(p ∨ q) → (p ∨ r)].
Write the following argument in symbolic form. Then either establish the validity of the argument or provide a counterexample to show that it is invalid. If it is cool this Friday, then Craig will wear his suede jacket if the pockets are mended. The forecast for Friday calls for cool weather, but
Consider the open statement p(x, y): y - x = y + x2 where the universe for each of the variables x, y comprises all integers. Determine the truth value for each of the following statements. (a) p(0, 0) (b) p(1, 1) (c) p(0, 1) (d) ∀y p(0, y) (e) ∃y p(1, y) (f) ∀x ∃y p(x, y) (g) ∃y ∀x
Determine whether each of the following statements is true or false. If false, provide a counterexample. The universe comprises all integers. (a) ∀x ∃y ∃z (x = 7y + 5z) (b) ∀x ∃y ∃z (x = 4y + 6z)
Suppose two opposite comer squares are removed from an 8 Ã— 8 chessboard - as in part (a) of Fig. 2.4. Can the remaining 62 squares be covered by 31 dominos (rectangles consisting of two adjacent squares - one white and the other blue, as shown in the figure)? (When a domino is placed
In part (b) of Fig. 2.4 we have an 8 Ã— 8 chessboard where two squares (one blue and one white) have been removed from each of two opposite comers. Can the remaining 60 squares be covered by 15 T-shaped figures (of three white squares and one blue one, or three blue squares and one
(a) Construct the truth table for (p → q) ∧ (¬p → r). (b) Translate the statement in part (a) into words such that the word "not" does not appear in the translation.
Let p, q, and r denote primitive statements. Prove or disprove (provide a counterexample for) each of the following. (a) [p ↔ (q ↔ r)] ⇔ [(p ↔ q) ↔ r] (b) [p → (q → r)] ⇔ [(p → q) → r]
Express the negation of the statement p ↔ q in terms of the connectives ∧ and ∨.
Write the following statement as an implication in two ways, each in the if-then form: Either Kaylyn practices her piano lessons or she will not go to the movies.
Let p, q, r denote primitive statements. Write the converse, inverse, and contrapositive of (a) p → (q ∧ r) (b) (p ∨ q) → r
(a) For primitive statements p, q, find the dual of the statement (¬p ∧ ¬q) ∨ (T0 ∧ p) ∨ p. (b) Use the laws of logic to show that your result from part (a) is logically equivalent to p ∧ ¬q.
Let p, q, r, and s be primitive statements. Write the dual of each of the following compound statements. (a) (p ∨ ¬q) ∧ (¬r ∨ s) (b) p → (q ∧ ¬r ∧ s) (c) [(p ∨ T0) ∧ (q ∨ F0)] ∨ [r ∧ s ∧ T0]

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