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- r is equivalent to either q or p Statememt-2 - r is equivalent to p - q A- Statement -1 is false- Statement -2 is trueB- Statement -1 is true- Statement -2 is true- Statement -2 is not a correct explanation for Statement -1C- Statement -1 is true- Statement -2 is true- Statement -2 is a correct explanation for Statement -1D- Statement -1 is true- Statement -2 is false

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Direct method of Validation

Let p be the ...

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 : $r i s e q u i v a l e n t t o e i t h e r q o r p$
Statememt-2 : $r i s e q u i v a l e n t t o (p \leftrightarrow\sim q)$

Statement – 1 is false, Statement – 2 is true

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Statement – 1 is true, Statement – 2 is true; Statement – 2 is a correct explanation for Statement -1

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Statement – 1 is true, Statement – 2 is true; Statement -2 is not a correct explanation for Statement -1

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Statement – 1 is true. Statement – 2 is false

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Solution

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 $(p \leftrightarrow\sim q)$

$\begin{matrix} 1 & 2 & 3 p & q & \sim p & \sim q & \sim p \leftrightarrow q & q o r p & p \leftrightarrow\sim q & \sim (p \leftrightarrow\sim q) T & T & F & F & F & T & F & T T & F & F & T & T & T & T & F F & T & T & F & T & T & T & F F & F & T & T & F & F & F & T \end{matrix}$
From columns (1) and (2), we observe, $r \equiv (p \leftrightarrow\sim q)$

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

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Similar questions

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 : $r i s e q u i v a l e n t t o e i t h e r q o r p$
Statememt-2 : $r i s e q u i v a l e n t t o (p \leftrightarrow\sim q)$

Q. Statement - 1:

\sim (p \leftrightarrow\sim q)

is equivalent to

p \leftrightarrow q

.
Statement - 2:

\sim (p \leftrightarrow\sim q)

is a tautology.

Statement 1: $\sim (p \leftrightarrow\sim q)$ is equivalent to $p \leftrightarrow q$
Statement 2: $\sim (p \leftrightarrow\sim q)$ is tautology.
Then which of the following statements is correct

Q. Statement - 1 : The statement

(p \lor q) \land \sim p

and

\sim p \land q

are logically equivalent.
Statement - 2 : The end columns of the truth table of both statements are identical.

Q. 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)

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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 : r is equivalent to either q or p Statememt-2 : r is equivalent to (p↔∼q)