Consider the statement P: For every real number x, if x^2 is greater than 0, then x is not equal to 0. Which of the following is the negation of P?

A. There exists a real number x such that x^2 > 0 and x = 0. — The negation of an implication p implies q is p and not q.

B. There exists a real number x such that x^2 0 and x 0.

C. For every real number x, x^2 > 0 and x = 0.

D. There exists a real number x such that x^2 > 0 or x = 0.

JEE Advanced · MathematicsMedium

$Consider the statement P : For every real number x, if x^{2} > 0, then x \neq = 0 . Which of the following is the negation of P ?$

Show correct answer & step-by-step solution

Correct answer: A — $There exists a real number x such that x^{2} > 0 and x = 0 .$

$The statement P is of the form \forall x, (p (x) ⟹ q (x)) where p (x) is x^{2} > 0 and q (x) is x \neq = 0 .$
$The negation of \forall x, (p (x) ⟹ q (x)) is \exists x, \neg (p (x) ⟹ q (x)) .$
$Since \neg (p ⟹ q) is equivalent to p \land \neg q, the negation is \exists x, (p (x) \land \neg q (x)) .$
$Substituting the expressions, we get \exists x, (x^{2} > 0 \land x = 0) .$
$Hence the answer is (A).$