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Chapter 3: Problem 4

Use a truth table to determine whether the two statements are equivalent. \(\sim p \rightarrow q, p \rightarrow \sim q\)

Short Answer

Expert verified
After performing the truth table operations, we find that for all combinations of truth values for \(p\) and \(q\), the resulting values for \(\sim p \rightarrow q\) and \(p \rightarrow \sim q\) are not the same. Therefore, the two logical statements are not equivalent.

Step by step solution

01

List all possible combinations of truth values for p and q

Given that each of the propositions \(p\) and \(q\) can either be True (T) or False (F), we end up with four possible combinations: TT, TF, FT, FF.
02

Construct the truth table

We build a table with columns labeled as \(p\), \(q\), \(\sim p\), \(\sim q\), \(\sim p \rightarrow q\) and \(p \rightarrow \sim q\). Fill the first two columns with the combinations from Step 1. To compute \(\sim p\) and \(\sim q\), simply reverse the truth values of \(p\) and \(q\), respectively. For \(\sim p \rightarrow q\) and \(p \rightarrow \sim q\), recall that the implication operator (鈫�) is False only when the first part is True and the second part is False; otherwise, it鈥檚 True.
03

Compare the two statements

When each row of the truth values for \(\sim p \rightarrow q\) and \(p \rightarrow \sim q\) is compared, if they are not the same, then the two statements are not equivalent. If they are the same for all combinations, then the statements are equivalent.

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