Question

Let \(*, \square \in \{\wedge, \vee\}\) be such that the Boolean expression \((p * \sim q) \implies (p \square q)\) is a tautology. Then :

• A. \(* = \wedge, \square = \wedge\)
• B. \(* = \wedge, \square = \vee\)
• C. \(* = \vee, \square = \wedge\)
• D. \(* = \vee, \square = \vee\)

Hint:

For an implication to be a tautology, any combination of variables that makes the antecedent (\(X\)) true must also make the consequent (\(Y\)) true. Checking critical cases is often faster than a full table.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept: 
A Boolean expression is a tautology if it is true for all possible truth values of its variables. For an implication \(X \implies Y\) to be a tautology, whenever \(X\) is true, \(Y\) must also be true. 
Step 2: Detailed Explanation: 
Let's test Option (B): \(* = \wedge, \square = \vee\). 
The expression is \((p \wedge \sim q) \implies (p \vee q)\). 
Truth Table: 

Since the last column is all True, the expression is a tautology for these operators. 
If we checked (A): \((p \wedge \sim q) \implies (p \wedge q)\). If \(p=T, q=F\), the LHS is \(T\) but RHS is \(F\), so it's not a tautology. 
Step 3: Final Answer: 
The operators are \(* = \wedge\) and \(\square = \vee\).