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 a rational number iff y is a transcendental number.
Statement-1: r is equivalent to either q or p.
Statement-2: r is equivalent to $\sim (p\leftrightarrow \sim q)$

• A. Statement-1 is true, Statement-2 is true; Statement -2 is not a correct explanation for Statement-1.
• B. Statement-1 is true, Statement-2 is false.
• C. Statement-1 is false, Statement-2 is true.
• D. Both statement 1 and 2 are false.

Answer 1

The correct option is D Both statement 1 and 2 are false.

p : x is an irrational number
q : y is a transcendental number
r : x is a rational number iff y is a transcendental number
$\Rightarrow r:\sim p\leftrightarrow q$
$s_1$ : q or p
$s_2$ : $\sim (p\leftrightarrow \sim q)$
Clearly $s_1$ and r are not equivalent $\Rightarrow$ Statement-1 is false.
Also $s_2$ and r are not equivalent
$\Rightarrow$ Statement-2 is also false. Hence none of the statement is correct.