Exercise 561Logic, sets and proofs › Quantifiers and negationHigh schoolAdded 3 Nov 2026

Negating a universal statement

#negation#universal#forall

Statement

Negate the statement: $\forall x\in\mathbb{R},\ x^2>0$ . Is the negation true?

Hint

$\neg(\forall x,\ P(x))\equiv\exists x,\ \neg P(x)$ .

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Negation. $\exists x\in\mathbb{R},\ x^2\le0$ .

Truth value of negation. True: $x=0$ gives $x^2=0\le0$ .

Conclusion. The original statement is false; its negation is true.