Question-1

logic

DA-2024

Let \(x\) and \(y\) be two propositions. Which of the following statements are tautologies?

\((\neg x \land y) \implies (y \implies x)\)
\((x \land \neg y) \implies (\neg x \implies y)\)
\((\neg x \land y) \implies (\neg x \implies y)\)
\((x \land \neg y) \implies (y \implies x)\)

Answer

\((\neg x \land y) \implies (y \implies x)\)
\((x \land \neg y) \implies (\neg x \implies y)\)
\((\neg x \land y) \implies (\neg x \implies y)\)
\((x \land \neg y) \implies (y \implies x)\)

Solution

Let us view each of these options as \(P \implies Q\). For this to be a tautology, \(Q\) should be true whenenever \(P\) is true. So we need to look for those values of \(x, y\) for which \(P\) is true.

\((\neg x \land y) \implies (y \implies x)\)
- \(P\) is true when \(x\) is false and \(y\) is true.
- \(Q\) would be false.
- Thus \(P \implies Q\) is false.
- It follows that \(P \implies Q\) is not a tautology.
\((x \land \neg y) \implies (\neg x \implies y)\)
- \(P\) is true when \(x\) is true and \(y\) is false.
- \(Q\) would be true.
- Thus \(P \implies Q\) is true.
- It follows that \(P \implies Q\) is a tautology.
A similar argument holds for the remaining options.