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Choose the correct answer. The statement $p \rightarrow (q \rightarrow p)$ is equivalent to which of the following?

$\begin{array}{1 1} (A) p \rightarrow (p \rightarrow q) \\(B) p \rightarrow (p \vee q) \\ (C) p \rightarrow (p \wedge q) \\ (D) p \leftrightarrow (p \rightarrow q)\end{array}$

1 Answer

$p \rightarrow (q \rightarrow p) \equiv \sim p \vee (q \rightarrow p)$ $\equiv \sim p \vee (\sim q \vee p)$
$\qquad \equiv \sim q \vee (p \vee \sim p ) \equiv \sim q \vee T = T$
$\Rightarrow p \rightarrow (q \rightarrow p) $ is a tautology.
Also, $p \rightarrow (p\vee q) \equiv \sim p \vee (p \vee q ) \equiv (~p \vee p) \vee q \equiv T \vee q \equiv T$
$\Rightarrow p \rightarrow (p \vee q ) $ is also a tautology.
$\Rightarrow p \rightarrow (q \rightarrow p) \equiv p \rightarrow (p \vee q ) $
answered Mar 20, 2014 by balaji.thirumalai

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