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 if and only if y is a transcendental number”.
Statement-1 : $risequivalenttoeitherqorp$
Statememt-2 : $risequivalentto(p\leftrightarrow \sim q)$

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

Answer 1

The correct option is A Statement – 1 is false, Statement – 2 is true

p : x is an irrational number
q : y is a transcendental number
r : x is a rational number iff y is a transcendental number.
Clearly r: $\sim p\leftrightarrow q$
Let us use truth table to check the equivalence of 'r' and 'q or p'; 'r' and $(p\leftrightarrow \sim q)$

$\begin{array}{cccccccc}& & & & 1& 2& & 3\\ p& q& \sim p& \sim q& \sim p\leftrightarrow q& qorp& p\leftrightarrow \sim q& \sim (p\leftrightarrow \sim q)\\ T& T& F& F& F& T& F& T\\ T& F& F& T& T& T& T& F\\ F& T& T& F& T& T& T& F\\ F& F& T& T& F& F& F& T\end{array}$

From columns (1) and (2), we observe,$r\equiv (p\leftrightarrow \sim q)$

$\therefore$ Statement-1 is false and statement-2 is true.
Option (a) is correct.