/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 Show that each of the following ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 11

Show that each of the following arguments is invalid by providing a counterexamplethat is, an assignment of truth values for the given primitive statements \(p, q, r\), and \(s\) such that all premises are true (have the truth value 1) while the conclusion is false (has the truth value 0 ). a) \([(p \wedge \neg q) \wedge[p \rightarrow(q \rightarrow r)]] \rightarrow \neg r\) b) \([[(p \wedge q) \rightarrow r] \wedge(\neg q \vee r)] \rightarrow p\) c) \(p \leftrightarrow q\) d) \(p\) \(q \rightarrow r\) \(p \rightarrow r\) \(r \vee \neg s\) \(p \rightarrow(q \vee \neg r)\) \(\frac{7 s \rightarrow q}{\therefore s}\)

Short Answer

Expert verified
What is being demonstrated in this exercise are the invalidities of the four sets of logical statements. A counterexample for each argument has been provided by assigning truth values to variables so that all premises are true while the conclusion is false.

Step by step solution

01

Part A

Consider the logical statement \( [(p \wedge \neg q) \wedge [p \rightarrow (q \rightarrow r)]] \rightarrow \neg r\). Let's assign the truth values as follows: \({p = true, q = false, r = true}\). This makes the overall statement false since the premises are true and the conclusion is false, hence invalid.
02

Part B

For the logical statement \([[(p \wedge q) \rightarrow r] \wedge (\neg q \vee r)] \rightarrow p\), decide on the truth values as follows: \({p = false, q = true, r = true}\). Again, with this assignment, all premises are true and the conclusion is false, hence proving its invalidity.
03

Part C

The statement here is \(p \leftrightarrow q\). If \(p =true\) and \(q =false\), then the statement is false, hence invalid.
04

Part D

For the last set of statements, adopt truthful values as shown: \(p = true, q = false, r = false, s = true\). Even with all premises true, the conclusion \(s\) is false, thereby illustrating the argument's invalidity.

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Most popular questions from this chapter

Let \(p, q, r\), and \(s\) be primitive statements. Write the dual of each of the following compound statements. a) \((p \vee \neg q) \wedge(\neg r \vee s)\) b) \(p \rightarrow(q \wedge \neg r \wedge s)\) c) \(\left[\left(p \vee T_{0}\right) \wedge\left(q \vee F_{0}\right)\right] \vee\left[r \wedge s \wedge T_{e}\right]\)

For the universe of all integers, let \(p(x), q(x), r(x), s(x)\), and \(t(x)\) be the following open statements. $$ \begin{array}{ll} p(x): & x>0 \\ q(x): & x \text { is even } \\ r(x): & x \text { is a perfect square } \\ s(x): & x \text { is (exactly) divisible by } 4 \\ t(x): & x \text { is (exactly) divisible by } 5 \end{array} $$ a) Write the following statements in symbolic form. i) At least one integer is even. ii) There exists a positive integer that is even. iii) If \(x\) is even, then \(x\) is not divisible by \(5 .\) iv) No even integer is divisible by \(5 .\) v) There exists an even integer divisible by \(5 .\) vi) If \(x\) is even and \(x\) is a perfect square, then \(x\) is divisible by \(4 .\) b) Determine whether each of the six statements in part (a) is true or false. For each false statement, provide a counterexample. e) Express each of the following symbolic representations in words. i) \(\forall x[r(x) \rightarrow p(x)]\) ii) \(\forall x[s(x) \rightarrow q(x)]\) iii) \(\forall x[s(x) \rightarrow \neg t(x)]\) iv) \(\exists x[s(x) \wedge \neg r(x)]\) v) \(\forall x[\neg r(x) \vee \neg q(x) \vee s(x)]\) d) Provide a counterexample for each false statement in part (c).

Give the reasons for the steps verifying the following argument. $$ \begin{aligned} &(\neg p \vee q) \rightarrow r \\ &r \rightarrow(s \vee t) \\ &\neg s \wedge \neg u \\ &\frac{\neg u \rightarrow \neg t}{\therefore p} \end{aligned} $$ Steps Reasons 1) \(\neg s \wedge \neg u\) 2) \(7 u\) 3) \(\neg u \rightarrow \neg t\) 4) \(\neg t\) 5) \(7 s\) 6) \(\neg s \wedge \neg t\) 7) \(r \rightarrow(s \vee t)\) 8) \(\neg(s \vee t) \rightarrow \neg r\) 9) \((\neg s \wedge \neg t) \rightarrow \neg r\) 10) \(\neg r\) 11) \((\neg p \vee q) \rightarrow r\) 12) \(\neg r \rightarrow \neg(\neg p \vee q)\) 13) \(\neg r \rightarrow(p \wedge \neg q)\) 14) \(p \wedge \neg q\) 15) \(\therefore p\)

Give the reason(s) for each step needed to show that the following argument is valid. $$ [p \wedge(p \rightarrow q) \wedge(s \vee r) \wedge(r \rightarrow \neg q)] \rightarrow(s \vee t) $$ Steps Reasons 1) \(p\) 2) \(p \rightarrow q\) 3) \(q\) 4) \(r \rightarrow \neg q\) 5) \(q \rightarrow \neg r\) 6) \({ }^{n} r\) 7) \(s \vee r\) 8) \(s\) 9) \(\therefore s \vee t\)

Let \(p, q, r\) denote the following statements about a particular triangle \(A B C, p:\) Triangle \(A B C\) is isosceles; \(q\) : Triangle \(A B C\) is equilateral; \(r\); Triangle \(A B C\) is equiangularTranslate each of the following into an English sentence. a) \(q \rightarrow p\) b) \(\neg p \rightarrow \neg q\) c) \(q \leftrightarrow r\) d) \(p \wedge \neg q\) e) \(r \rightarrow p\)
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