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# Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number” and r be the statement “x is a rational number iff y is a transcendental number”.  S1 : r is equivalent to either q or p  S2 : r is equivalent to $^{\sim}(p \leftrightarrow \: ^{\sim}q)$

$\begin {array} {1 1} \text{(A) S1 is true, S2 is false} \\ \text{(B) S1 is false,S2 is true} \\ \text{(C) both S1 and S2 are true} \\ \text{(D) both S1 and S2 are false} \end {array}$

 p p $^{\sim}p$ $^{\sim}q$ $^{\sim}P \leftrightarrow q$ q V p $P \leftrightarrow ^{\sim}q$ $^{\sim}(P \leftrightarrow ^{\sim}q)$ T T F F F T F T T F F T T T T F F T T F T T T F F F T T F F F T
Ans : (D)
Therefore from the given statements p,q, and r,
$r: \: ^{\sim}p \leftrightarrow q$
So, $S1 : r \approx q\: V\: p \: and \: S2 : r \approx\: ^{\sim}(p \leftrightarrow ^{\sim}q)$
So, it is clear from the table that r is not equivalent to either of the statements.

edited Jan 22, 2014