Question

If the Boolean expression \((p \Rightarrow q) \Leftrightarrow (q * (\sim p))\) is a tautology, then the boolean expression \((p * (\sim q))\) is equivalent to :

• A. \(q \Rightarrow p\)
• B. \(p \Rightarrow q\)
• C. \(p \Rightarrow \sim q\)
• D. \(\sim q \Rightarrow p\)

Hint:

Since \(p \Rightarrow q \equiv \sim p \vee q\), the connective \(*\) must be \(\vee\). Then \(p \vee \sim q\) is \(q \Rightarrow p\).

Answer 1

The Correct Option is A

We know \(p \Rightarrow q \equiv (\sim p) \vee q\). For \((p \Rightarrow q) \Leftrightarrow (q * \sim p)\) to be a tautology, \(q * \sim p\) must be equivalent to \(q \vee \sim p\), so \(* = \vee\).
(Check with \(*= \wedge\): for \(p = T, q = T\), \(p \Rightarrow q = T\) but \(q \wedge \sim p = F\), so it fails.)
Then \(p * (\sim q) = p \vee (\sim q) = (\sim q) \vee p \equiv q \Rightarrow p\).
Answer: (1) \(q \Rightarrow p\)