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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".
S - I : r is equivalent to either q or p. S - II : r is equivalent to $~ (p \leftrightarrow\sim q)$

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S - I is true, S - II is true, S - II is a correct explanation of S - I

S - I is true, S - II is true, S - II is not a correct explanation of S - I

S - I is true, S - II is false

S - I is false, S - II is false

answer is D.

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Detailed Solution

$r :\sim p \leftrightarrow q,$

$S_{1} : r \equiv q \lor p, S_{2} : r \equiv\sim (p \leftrightarrow\sim q)$
using truth tables both S₁ and S₂ are false

p	q	$~ p$	$~ p \leftrightarrow q$	$q \lor p$
T	T	F	F	T
T	F	F	T	T
F	T	T	T	T
F	F	T	F	F

p	q	$~ p$	$~ q$	$p \leftrightarrow\sim q$	$~ (p \leftrightarrow\sim q)$	$~ p \leftrightarrow q$
T	T	F	F	F	T	F
T	F	F	T	T	F	T
F	T	T	F	T	F	T
F	F	T	T	F	T	F

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