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 iff y is transcendental number’.
Statement 1 :- r is equivalent to either q or p
Statement 2 :- r is equivalent to $\sim (p\leftrightarrow \sim q)$$\sim (p\leftrightarrow \sim q)$
(a) Statement 1 is correct, statement 2 is correct; statement 2 is the correct explanation for statement 1
(b) Statement 1 is correct, statement 2 is correct; statement 2 is not the correct explanation for statement 1
( c) Statement 1 is correct, statement 2 is incorrect
(d) Statement 1 is correct, statement 2 is correct
(e) None of these

Answer 1

Hint: We have given the statements and we have to find out which of the statements is true. To solve the question, we use the truth table to get full clarification about the statements and find out which of the statements is true.

Step-by- step explanation:
Given P :- x is an irrational number.
Q :- y is a transcendental number.
R :- x is a rational number if y is a transcendental number
We have to show which statement is true.
From the given statements, r: $\sim (p\leftrightarrow q)$
${{s}_{1}}$ : q or p
${{s}_{2}}$ : $\sim (p\leftrightarrow q)$
Let us use the truth table to check the equivalence of r and q or p and $\sim (p\leftrightarrow q)$

p	q	$\sim p$	$\sim q$	$\sim (p\leftrightarrow q)$	Q or p	$p\leftrightarrow \sim q$	$\sim (p\leftrightarrow \sim q)$
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

From columns, 1, 2 and 3 , we observe that none of these statements are equivalent to each other.
${{s}_{1}}$and r are not equivalent
This means statement 1 is false.
${{s}_{2}}$and r are not equivalent.
This means that statement 2 is also false.
None of the statements are true.
Option (E) is correct.

Note: Students make mistakes in making the truth table. To make the truth table, we must have the proper knowledge of the symbols then we will be able to make the truth table and find out the right answer.