Let \(x\) and \(y\) be two propositions. Which of the following statements is a tautology /are tautologies?
Answer
(B), (C), (D)
Explanation
An implication is false only when its premise is true and its conclusion is false. Each premise here is true in exactly one row of the truth table, so check the conclusion in that row.
- A: \(\neg x \land y\) needs \(x\) false and \(y\) true. Then \(y \Longrightarrow x\) is true \(\Longrightarrow\) false, which is false. Not a tautology.
- B: \(x \land \neg y\) needs \(x\) true and \(y\) false. Then \(\neg x\) is false, so \(\neg x \Longrightarrow y\) is true. Tautology.
- C: \(x\) false and \(y\) true make \(\neg x \Longrightarrow y\) true \(\Longrightarrow\) true, which is true. Tautology.
- D: \(x\) true and \(y\) false make \(y \Longrightarrow x\) false \(\Longrightarrow\) true, which is true. Tautology.
Answer: B, C, D