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Chapter 3: Problem 34

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

Short Answer

Expert verified
The logical statement \([(p \rightarrow q) \wedge p] \rightarrow q\) is a tautology because the final column of the truth table is always True.

Step by step solution

01

Construct the Truth Values of p and q

The first step is to construct a table with all possible truth values of p and q. Since p and q are boolean variables, each of them can be either True (T) or False (F). Thus we have 4 combinations in total: TT, TF, FT, FF.
02

Calculate Truth Values for \(p \rightarrow q\)

Now, calculate the truth values for the implication \(p \rightarrow q\). The rule is: it's True except for the case where p is True and q is False.
03

Calculate Truth Values for \([(p \rightarrow q) \wedge p]\)

Using the values from steps 1 and 2, calculate the truth values for the compound statement \([(p \rightarrow q) \wedge p]\). The logical operator \(\wedge\) results in True when both operands are True.
04

Complete the Truth Table for the Given Statement

Finally, compute the truth values of the whole given statement \([(p \rightarrow q) \wedge p] \rightarrow q\) by referring to the previously computed columns in the truth table.
05

Analyze the Truth Values

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

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