Question

Let the operations $*, \odot \in\{\wedge, \vee\}$. If $(p * q) \odot(p \odot \sim q)$ is a tautology, then the ordered pair $(*, \odot)$ is :

• A. $(\vee, \wedge)$
• B. $(\vee, \vee)$
• C. $(\wedge, \wedge)$
• D. $(\wedge, \vee)$

Answer 1

The Correct Option is B

 To determine the correct ordered pair \( (*, \odot) \) for which the expression \((p * q) \odot(p \odot \sim q)\) is a tautology, we need to analyze the possible operations for \(*\) and \(\odot\) from the given set \(\{\wedge, \vee\}\).

Note that:

• \(\wedge\) represents the logical AND.
• \(\vee\) represents the logical OR.

Let's evaluate the expression for each pair by substitution:

1. For \((*, \odot) = (\vee, \wedge)\): The expression becomes \( (p \vee q) \wedge (p \wedge \sim q) \).
   • Consider \( p = \text{True} \) and \( q = \text{True} \):
   • \( p \vee q = \text{True} \), \( \sim q = \text{False} \), so \( p \wedge \sim q = \text{False} \).
   • \( (p \vee q) \wedge (p \wedge \sim q) = \text{True} \wedge \text{False} = \text{False} \).
2. For \((*, \odot) = (\vee, \vee)\): The expression becomes \( (p \vee q) \vee (p \vee \sim q) \).
   • This is always \(\text{True}\), because at least one of \(p \vee q\) or \(p \vee \sim q\) is \(\text{True}\) for all combinations of \( p \) and \( q \).
3. For \((*, \odot) = (\wedge, \wedge)\): The expression becomes \( (p \wedge q) \wedge (p \wedge \sim q) \).
   • Consider \( p = \text{True} \) and \( q = \text{True} \):
   • \( p \wedge q = \text{True} \); however, with \( \sim q = \text{False}, p \wedge \sim q = \text{False} \).
   • \( (p \wedge q) \wedge (p \wedge \sim q) = \text{True} \wedge \text{False} = \text{False} \).
4. For \((*, \odot) = (\wedge, \vee)\): The expression becomes \( (p \wedge q) \vee (p \vee \sim q) \).
   • Consider \( p = \text{False} \) and \( q = \text{True} \):
   • \( p \wedge q = \text{False} \) and \( p \vee \sim q = \text{False} \). Hence, the whole expression is \(\text{False} \).

From the above evaluations, the only pair that makes the expression a tautology is \((\vee, \vee)\). Thus, the correct ordered pair is:

Answer: \((\vee, \vee)\)