MHTCET 11 May 2024 Shift 2 Paper

Concept:A logical statement that is true for all possible truth values is called a tautology.Explanation:Start with the given logical expression:$$(p \rightarrow q) \rightarrow ((\sim p \rightarrow q) \rightarrow q)$$Use the identity $p \rightarrow q \equiv \sim p \vee q$.Replace the inner implication $\sim p \rightarrow q$:$$\sim p \rightarrow q \equiv p \vee q$$Now the expression becomes:$$(p \rightarrow q) \rightarrow ((p \vee q) \rightarrow q)$$Simplify $(p \vee q) \rightarrow q$ using $A \rightarrow B \equiv \sim A \vee B$:$$(p \vee q) \rightarrow q \equiv \sim(p \vee q) \vee q$$$$\equiv (\sim p \wedge \sim q) \vee q$$Use the distributive law:$$\equiv (\sim p \vee q) \wedge (\sim q \vee q)$$Since $\sim q \vee q \equiv T$, we get:$$\equiv (\sim p \vee q) \wedge T$$$$\equiv \sim p \vee q$$But $\sim p \vee q \equiv p \rightarrow q$.Therefore, the whole expression simplifies to:$$(p \rightarrow q) \rightarrow (p \rightarrow q) \equiv T$$So the given statement is always true.Answer:D. a tautology.