Exercise 504Logic, sets and proofs › Proof techniquesHigh schoolAdded 3 Nov 2026

Negating quantifiers

#quantifiers#negation#logic

Statement

Write the negation of: " $\forall x\in\mathbb{R}$ , $\exists y\in\mathbb{R}$ such that $x+y>0$ ."

Hint

Swap $\forall\leftrightarrow\exists$ and negate the inner statement.

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Negation.
$\exists x\in\mathbb{R},\ \forall y\in\mathbb{R},\ x+y\le0.$

Interpretation. There exists some $x$ such that for all $y$ , $x+y\le0$ . (This is false, since $y$ can be chosen large.)

Conclusion. The negation is $\exists x\in\mathbb{R},\forall y\in\mathbb{R},\ x+y\le0$ .