# Statement-1: $(p \wedge \sim q) \wedge (\sim p \wedge q)$ is a fallacy, and Statement-2 : $(p \rightarrow q) \leftrightarrow (\sim q \rightarrow \sim p)$ is a tautology. Given these two statements, which of the following is true?

(A) Statement-1 is true, Statement-2 is false. (B) Statement-1 is false, Statement-2 is true. (C) Both statements are true, but Statement-2 is not a correct explanation of Statement-1 (D) Statement-1 is true, and Statement-2 is a correct explanation of Statement-1

## 1 Answer

Statement-1: $(p \wedge \sim q) \wedge (\sim p \wedge q)$ is a fallacy.

Statement-2 : $(p \rightarrow q) \leftrightarrow (\sim q \rightarrow \sim p)$

Truth table for Statement-1: $(p \wedge \sim q) \wedge (\sim p \wedge q)$:
$\begin{matrix} p &q & \sim p & \sim q & (p \wedge \sim q) & (\sim p \wedge q) & (p \wedge \sim q ) \wedge (\sim p \wedge q)\\ T&T &F &F & F &F &F \\ T& F & F & T & T&F & F\\ F& T& T& F& F &T & F\\ F& F& T& T& F&F & F \end{matrix}$
This is a fallacy.
Truth table for Statement-2 : $(p \rightarrow q) \leftrightarrow (\sim q \rightarrow \sim p)$:
$\begin{matrix} p & q & \sim p &\sim q & (p \rightarrow q) & (\sim q \rightarrow \sim p) & (p \rightarrow q )\leftrightarrow ( \sim q \rightarrow \sim p)\\ T&T &F &F & T &T &T \\ T& F & F & T & F&F & T\\ F& T& T& F& T&T & T\\ F& F& T& T& T& T & T \end{matrix}$
This is a tautology.
Therefore, while both statements are true, we can see that Statement-2 is not an explanation of Statement-1.
answered Mar 20, 2014

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