Which of the following Boolean expression is a tautology ?
Show Hint
- \( p \Leftrightarrow p \vee q \)
- \( (p \wedge q) \Leftrightarrow p \)
- \( (p \wedge(\sim p \vee q)) \vee(\sim q) \)
- \( (p \vee \sim p) \wedge(q \vee(\sim q)) \)
The Correct Option is D
Solution and Explanation
(1) \( p\Leftrightarrow (p\vee q) \): take p = F, q = T; then LHS = F and RHS = T, so it is false. Not a tautology.
(2) \( (p\wedge q)\Leftrightarrow p \): take p = T, q = F; LHS = F, RHS = T, so false. Not a tautology.
(3) \( (p\wedge(\sim p\vee q))\vee\sim q = (p\wedge q)\vee\sim q = p\vee\sim q \) (absorbing). For p = F, q = T it is false. Not a tautology.
(4) \( p\vee\sim p \) is always T and \( q\vee\sim q \) is always T, so their conjunction is always T. This is a tautology.
Answer: (4) \( (p \vee \sim p) \wedge(q \vee(\sim q)) \)
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