Question

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 rational number if $y$ is transcendental number'. Statement I: $r$ is equivalent to either $q$ or $p$. Statement II: $r$ is equivalent to $~(p\leftrightarrow ~q)$.

• A. Statement I is correct, Statement II is correct; Statement II is the correct explanation for Statement I.
• B. Statement I is correct, Statement II is correct; Statement II is not a correct explanation for Statement I.
• C. Statement I is correct, Statement II is incorrect
• D. Statement I is incorrect, Statement II is correct.
• E. None of these.

Answer 1

The correct option is E

None of these.

Explanation for the correct option:

$p$: $x$ is an irrational number.

$q$: $y$ is a transcendental number.

$r$: $x$ is a rational number iff $y$ is a transcendental number.

$\Rightarrow r:~p\leftrightarrow q$

Now, let $s_1$ and $s_2$ be two statements such that

$s_1:q\vee p$

$s_2:~(p\leftrightarrow ~q)$

Here,

$s_1\not\equiv r$ and $s_2\not\equiv r$.

Therefore, both statements are incorrect.

Hence, the correct option is (E).