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Chapter 3: Problem 52

Use a truth table to determine whether each statement is a tautology, a self- contradiction, or neither. \((p \wedge q) \rightarrow(\sim q \vee r)\)

Short Answer

Expert verified
Whether the logical statement is a tautology, a self-contradiction, or neither can only be determined after completing the truth table and analyzing it. Without such analysis, providing an answer would be not feasible.

Step by step solution

01

Set up the truth table

Create a truth table with four columns, one each for \(p, q, r\), and another for the logical expression \((p \wedge q) \rightarrow(\sim q \vee r)\). Fill the table with all combinations of truth values for \(p, q\n) and \(r\), i.e., \(T,T,T; T,T,F; T,F,T; T,F,F; F,T,T; F,T,F; F,F,T; F,F,F\n) where T represents True and F represents False.
02

Compute intermediate values

Evaluate the truth values of the intermediate logical expressions in the given logical statement. Calculate the truth values for \(p \wedge q\), \(\sim q\), and \(\sim q \vee r\). Add these to the truth table.
03

Evaluate the entire logical expression

Using the truth values of the intermediate logical expressions, evaluate the truth values of the entire logical statement \((p \wedge q) \rightarrow(\sim q \vee r)\) for all combinations of truth values of \(p, q\n) and \(r\).
04

Determine if the statement is a tautology, self-contradiction, or neither

Analyze the truth values of the logical statement to determine whether it's a tautology, self-contradiction, or neither. If all rows in the column of the logical statement are true, it's a tautology. If all rows are false, it's a self-contradiction. If there are both true and false values, the statement is neither a self-contradiction nor a tautology.

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Most popular questions from this chapter

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