Question

Among the given statements below 
 (a) \( \neg p \vee (\neg p \vee q) \) 
 (b) \( \neg q \wedge (\neg p \vee \neg q) \) 
 (c) \( (\neg p \vee \neg q) \wedge (p \vee \neg q) \) 
 (d) \( (\neg p \vee \neg q) \vee (p \vee \neg q) \)
 .............. is a tautology.
 

• (b)
• (a)
• (c)
• (d)

Hint:

To determine if a logical statement is a tautology, construct the truth table for the expression and check if it evaluates to true for all possible combinations of truth values.

Answer 1

The Correct Option is D

Step 1: Analyze each logical statement.
We need to check whether any of the given logical statements result in a tautology. A tautology is a statement that is always true, regardless of the truth values of its components.
Step 2: Check the truth tables for each statement.
We will construct the truth tables for the given options and determine if any of the statements are always true. For statement (d): \[ (\neg p \vee \neg q) \vee (p \vee \neg q) \] This expression simplifies to: \[ (\neg p \vee \neg q) \vee (p \vee \neg q) = \text{true} \] This is a tautology because it evaluates to true for all possible truth values of \( p \) and \( q \).
Step 3: Conclusion.
Thus, the correct answer is (d), which is a tautology.