Which of the following Boolean expressions is not a tautology?
Show Hint
- \((p \Rightarrow q) \vee (\sim q \Rightarrow p)\)
- \((q \Rightarrow p) \vee (\sim q \Rightarrow p)\)
- \((p \Rightarrow \sim q) \vee (\sim q \Rightarrow p)\)
- \((\sim p \Rightarrow q) \vee (\sim q \Rightarrow p)\)
The Correct Option is D
Solution and Explanation
(1) \((p \Rightarrow q) \vee (\sim q \Rightarrow p) = (\sim p \vee q) \vee (q \vee p)\), which contains \(p \vee \sim p\), a tautology.
(2) \((q \Rightarrow p) \vee (\sim q \Rightarrow p) = (\sim q \vee p) \vee (q \vee p)\), which contains \(q \vee \sim q\), a tautology.
(3) \((p \Rightarrow \sim q) \vee (\sim q \Rightarrow p) = (\sim p \vee \sim q) \vee (q \vee p)\), which contains \(p \vee \sim p\), a tautology.
(4) \((\sim p \Rightarrow q) \vee (\sim q \Rightarrow p) = (p \vee q) \vee (q \vee p) = p \vee q\). This is false when \(p\) and \(q\) are both false, so it is not a tautology.
Answer: (4) \((\sim p \Rightarrow q) \vee (\sim q \Rightarrow p)\)
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