Let \(*, \square \in \{\wedge, \vee\}\) be such that the Boolean expression \((p * \sim q) \implies (p \square q)\) is a tautology. Then :
Show Hint
- \(* = \wedge, \square = \wedge\)
- \(* = \wedge, \square = \vee\)
- \(* = \vee, \square = \wedge\)
- \(* = \vee, \square = \vee\)
The Correct Option is B
Solution and Explanation
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 \(\square = \vee\).
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