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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.
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B
Statement-1 is true, Statement-2 is false.
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C
Statement-1 is false, Statement-2 is true.
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D
Both statement 1 and 2 are false.
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Solution

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.  
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Maths

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