Question

If the Boolean expression $ (p \Rightarrow q) \Leftrightarrow (q \land (\sim p)) $ is a tautology, then the Boolean expression $ (p \land (\sim q)) $ is equivalent to

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

Hint:

Tautologies often help us find relationships between variables. Here, simplifying the expression helps us deduce the correct equivalence for the given Boolean expression.

Answer 1

The Correct Option is B

We are given that the Boolean expression \( (p \Rightarrow q) \Leftrightarrow (q \land (\sim p)) \) is a tautology. This condition means that the truth values of both sides of the equivalence must always be the same. By simplifying the tautology, we deduce the relationship between \( p \) and \( q \). For the expression \( (p \land (\sim q)) \), it is equivalent to the logical implication \( q \Rightarrow p \), as this is the expression that satisfies the given tautology condition.
Therefore, the correct answer is 2. \( q \Rightarrow p \).