Question

If \(q\) is false and \(p \wedge q \leftrightarrow r\) is true, then which one of the following statements is a tautology ?

• A. \(p \wedge r\)
• B. \((p \vee r) \rightarrow (p \wedge r)\)
• C. \(p \vee r\)
• D. \((p \wedge r) \rightarrow (p \vee r)\)

Hint:

A tautology is true for every truth value of its variables. \((p \wedge r) \rightarrow (p \vee r)\) can never be false since \(p \wedge r\) true forces \(p \vee r\) true.

Answer 1

The Correct Option is D

Since \(q\) is false, \(p \wedge q\) is false. As \((p \wedge q) \leftrightarrow r\) is true, \(r\) must be false.
Now test each option with \(r = F\):
(1) \(p \wedge r = F\) for every \(p\): not a tautology.
(2) \((p \vee r) \rightarrow (p \wedge r)\) becomes \(p \rightarrow F\), which is false when \(p = T\): not a tautology.
(3) \(p \vee r = p\): not always true.
(4) \((p \wedge r) \rightarrow (p \vee r)\): the antecedent \(p \wedge r\) is false, so the implication is true for both values of \(p\); in fact it is true for all \(p, r\) because \(p \wedge r\) implies \(p \vee r\). It is a tautology.
Answer: (4) \((p \wedge r) \rightarrow (p \vee r)\)