GATE DS&AI 2024 | Question: 19

Question

​​​​​​Let $x$ and $y$ be two propositions. Which of the following statements is a tautology /are tautologies?

• A. $(\neg x \wedge y) \Rightarrow(y \Rightarrow x)$
• B. $(x \wedge \neg y) \Rightarrow(\neg x \Rightarrow y)$
• C. $(\neg x \wedge y) \Rightarrow(\neg x \Rightarrow y)$
• D. $(x \wedge \neg y) \Rightarrow(y \Rightarrow x)$

Answer 1

Checking by Elimination:

X -> Y is Fallacy (False) if X = T and Y = F.

Let's try to make RHS of propositions a Fallacy, and put those values of X and Y in LHS to check if an option is tautology.

A) for X = F, Y = T we get RHS = F an LHS = T.
Overall, T -> F  = F. Hence it is not tautology.

B) for X= F, Y = F we get F -> F = T. Tautology

C) for X= F, Y = F we get F -> F = T. Tautology

D) for X= F, Y = T we get F -> F = T. Tauotlogy

B, C, D are correct options.