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Chapter 3: Problem 53

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

Short Answer

Expert verified
The given statement \([(p \rightarrow q) \wedge (q \rightarrow r)] \rightarrow (p \rightarrow r)\) is a tautology.

Step by step solution

01

Setting Up the Truth Table

Start by setting up a truth table for all the propositions contained in the logical statement. Here, the propositions are \(p\), \(q\), and \(r\). Our truth table will have eight rows, corresponding to the eight possible combinations of true and false for the three propositions.
02

Fill in the Truth Table

Next, fill in the truth value for \((p \rightarrow q)\), \((q \rightarrow r)\), and \((p \rightarrow r)\) for each row. Remember that \(p \rightarrow q\) is false only when \(p\) is true and \(q\) is false; in all other cases, it is true. Do the same for \((q \rightarrow r)\) and \((p \rightarrow r)\). After that, calculate the truth value for \((p \rightarrow q) \wedge (q \rightarrow r)\). Recall that \((A) \wedge (B)\) is true only when both \(A\) and \(B\) are true.
03

Determine the Truth Value of the Entire Statement

Finally, calculate the truth value for the entire statement \([(p \rightarrow q) \wedge (q \rightarrow r)] \rightarrow (p \rightarrow r)\). The statement is true in all rows.
04

Conclusion

Given that the statement is true in all rows, it can therefore be concluded that the given statement is a tautology.

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