Which of the following propositions is a tautology?
$A. \ \ (p \vee q) \rightarrow p$ $\quad \quad \equiv \ \ \neg(p \vee q) \vee p$ $\quad \quad \equiv \ \ (\neg p \wedge \neg q) \vee p$ $\quad \quad \equiv \ \ (\neg p \vee p)\wedge(\neg q \vee p)$ $\quad \quad \equiv (p \vee\neg q)$ $B. \ \ p \vee (q \rightarrow p)$ $\quad \quad \equiv \ \ p \vee (\neg q \vee p)$ $\quad \quad \equiv \ \ (p \vee p)\vee\neg q$ $\quad \quad \equiv p \vee\neg q$ $C. \ \ p \vee (p \rightarrow q)$ $\quad \quad \equiv \ \ p \vee (\neg p \vee q)$ $\quad \quad \equiv \ \ ( p \vee \neg p) \vee q$ $\quad \quad \equiv \ \ T \vee q$ $\quad \quad \equiv \ \ T$ $D. \ \ p \rightarrow (p \rightarrow q)$ $\quad \quad \equiv \ \ p \rightarrow (\neg p \vee q)$ $\quad \quad \equiv \ \ \neg p \vee(\neg p \vee q)$ $\quad \quad \equiv \ \ (\neg p \vee\neg p) \vee q$ $\quad \quad \equiv \ \ \neg p \vee q$ Hence, Option(C) $p \vee (p \rightarrow q)$.
Gatecse
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