Tautology Statements Analysis in Propositional Logic

A tautology is a logical statement that is true regardless of the truth values of its individual propositions. To identify tautologies among the given options involving propositions $x$ and $y$, we simplify each expression using standard logical equivalences. Key rules include: $ P \Rightarrow Q \equiv \neg P \vee Q $, De Morgan's laws ($ \neg(P \wedge Q) \equiv \neg P \vee \neg Q $ and $ \neg(P \vee Q) \equiv \neg P \wedge \neg Q $), and double negation ($ \neg(\neg P) \equiv P $).

Option 1 Analysis: $ (\neg x \wedge y ) \Rightarrow (y \Rightarrow x) $

We simplify the expression using logical equivalences:

• $ (\neg x \wedge y ) \Rightarrow (y \Rightarrow x) $
• $ \equiv \neg(\neg x \wedge y ) \vee (y \Rightarrow x) $ (Using $ P \Rightarrow Q \equiv \neg P \vee Q $)
• $ \equiv (\neg(\neg x) \vee \neg y) \vee (\neg y \vee x) $ (Using De Morgan's law and $ P \Rightarrow Q \equiv \neg P \vee Q $)
• $ \equiv (x \vee \neg y) \vee (\neg y \vee x) $ (Using double negation $ \neg(\neg P) \equiv P $)
• $ \equiv x \vee \neg y \vee \neg y \vee x $ (Using associative and commutative laws)
• $ \equiv x \vee \neg y $ (Simplifying)

The resulting expression $ x \vee \neg y $ is not always true. For example, if $ x $ is False and $ y $ is True, the expression evaluates to False. Thus, Option 1 is not a tautology.

Option 2 Analysis: $ (x \wedge \neg y ) \Rightarrow (\neg x \Rightarrow y) $

We simplify the expression:

• $ (x \wedge \neg y ) \Rightarrow (\neg x \Rightarrow y) $
• $ \equiv \neg(x \wedge \neg y ) \vee (\neg x \Rightarrow y) $
• $ \equiv (\neg x \vee \neg(\neg y)) \vee (\neg(\neg x) \vee y) $
• $ \equiv (\neg x \vee y) \vee (x \vee y) $
• $ \equiv \neg x \vee y \vee x \vee y $
• $ \equiv (\neg x \vee x) \vee (y \vee y) $
• $ \equiv \text{True} \vee y $ (Since $ \neg P \vee P $ is always True)
• $ \equiv \text{True} $

The statement simplifies to True, confirming it is a tautology.

Option 3 Analysis: $ (\neg x \wedge y ) \Rightarrow (\neg x \Rightarrow y) $

We simplify the expression:

• $ (\neg x \wedge y ) \Rightarrow (\neg x \Rightarrow y) $
• $ \equiv \neg(\neg x \wedge y ) \vee (\neg x \Rightarrow y) $
• $ \equiv (\neg(\neg x) \vee \neg y) \vee (\neg(\neg x) \vee y) $
• $ \equiv (x \vee \neg y) \vee (x \vee y) $
• $ \equiv x \vee \neg y \vee x \vee y $
• $ \equiv (x \vee x) \vee (\neg y \vee y) $
• $ \equiv x \vee \text{True} $ (Since $ \neg P \vee P $ is always True)
• $ \equiv \text{True} $

The statement simplifies to True, confirming it is a tautology.

Option 4 Analysis: $ (x \wedge \neg y ) \Rightarrow (y \Rightarrow x) $

We simplify the expression:

• $ (x \wedge \neg y ) \Rightarrow (y \Rightarrow x) $
• $ \equiv \neg(x \wedge \neg y ) \vee (y \Rightarrow x) $
• $ \equiv (\neg x \vee \neg(\neg y)) \vee (\neg y \vee x) $
• $ \equiv (\neg x \vee y) \vee (\neg y \vee x) $
• $ \equiv \neg x \vee y \vee \neg y \vee x $
• $ \equiv (\neg x \vee x) \vee (y \vee \neg y) $
• $ \equiv \text{True} \vee \text{True} $ (Since $ P \vee \neg P $ and $ \neg P \vee P $ are always True)
• $ \equiv \text{True} $

The statement simplifies to True, confirming it is a tautology.

Tautologies Conclusion

After analyzing each statement using logical equivalences, we find that statements 2, 3, and 4 are tautologies because they logically simplify to True.