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Chapter 3: Problem 12

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

Short Answer

Expert verified
The two given statements are equivalent if and only if the outputs of \((p \wedge \sim r) \rightarrow q\) and \((\sim p \vee r) \rightarrow \sim q\), for all combinations of \(p\), \(q\), and \(r\) values, are identical in the truth table.

Step by step solution

01

Setting up the Truth Table

Start by setting up a truth table with columns for \(p\), \(q\), \(r\), \((p \wedge \sim r)\), \((p \wedge \sim r) \rightarrow q\), \((\sim p \vee r)\) and \((\sim p \vee r) \rightarrow \sim q\). Fill the columns of \(p\), \(q\), and \(r\) with all possible combinations of True (T) and False (F) values.
02

Calculating Expressions

Calculate the values of the expressions \((p \wedge \sim r)\) and \((\sim p \vee r)\) for all possible combinations, by substituting \(p\), \(q\), and \(r\) values. \(\wedge\) stands for AND, \(\vee\) is OR and \(\sim\) is NOT. AND operation returns True only if both the conditions are True. OR operation returns True if either or both of the conditions are True. NOT operation turns True into False and False into True.
03

Final Calculations

Substitute the calculated values of \((p \wedge \sim r)\) and \((\sim p \vee r)\) into the given statements \((p \wedge \sim r) \rightarrow q\) and \((\sim p \vee r) \rightarrow \sim q\), respectively. \(a \rightarrow b\) means 'if a then b', which is False only when \(a\) is True and \(b\) is False, in all other cases it is True.
04

Compare the Results

Check if the final results for the two expressions \((p \wedge \sim r) \rightarrow q\) and \((\sim p \vee r) \rightarrow \sim q\), for all combinations of \(p\), \(q\), and \(r\) values, are equal. If the results are identical for all cases, then the two given logical statements are said to be equivalent.

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