JEE Main · 2008 · 27 Apr 2008 · Offline · Mathematical Reasoning *

Writing $p$ for '$x$ is irrational' and $q$ for '$y$ is transcendental', the statement $r$, that $x$ is rational if and only if $y$ is transcendental, is $\sim p\leftrightarrow q$, true precisely when exactly one of $p,q$ holds, the exclusive or of $p$ and $q$; so the first statement, calling $r$ either $p$ or $q$, is true. The second offers $\sim (p\leftrightarrow \sim q)$, but $p\leftrightarrow \sim q$ is true when exactly one of $p,q$ holds, so its negation is true when both or neither hold, the opposite of $r$. Hence the first statement is true and the second is false.