Which of the following is a tautology?

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(A) To determine which expression is a tautology,we evaluate the truth table or simplify the logical expressions.For option $A$: $(\sim p \wedge (p \v

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$(\sim p) \wedge (p \vee q) \rightarrow q$

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(A) To determine which expression is a tautology,we evaluate the truth table or simplify the logical expressions.For option $A$: $(\sim p \wedge (p \vee q)) \rightarrow q$Using the distributive law: $(\sim p \wedge p) \vee (\sim p \wedge q) \rightarrow q$Since $(\sim p \wedge p) \equiv F$ (Contradiction),we have: $F \vee (\sim p \wedge q) \rightarrow q$This simplifies to: $(\sim p \wedge q) \rightarrow q$Using the implication law $a \rightarrow b \equiv \sim a \vee b$: $\sim (\sim p \wedge q) \vee q$Applying De Morgan's law: $(p \vee \sim q) \vee q$By associativity: $p \vee (\sim q \vee q)$Since $(\sim q \vee q) \equiv T$ (Tautology): $p \vee T \equiv T$Thus,option $A$ is a tautology.

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