Exercise 1162Logic, sets and proofs › Quantifiers and negationL1Added 3 Nov 2026

Order of quantifiers matters

#quantifier order#forall exists#difference

Statement

Compare the truth values of:
- $P_1$ : $\forall x\in\mathbb{R},\ \exists y\in\mathbb{R},\ y>x$ ;
- $P_2$ : $\exists y\in\mathbb{R},\ \forall x\in\mathbb{R},\ y>x$ .

Hint

In $P_1$ , $y$ may depend on $x$ ; in $P_2$ , one $y$ must work for all $x$ .

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Solution

$P_1$ is true. For each $x$ , choose $y=x+1$ .

$P_2$ is false. No real number is larger than every real.

Conclusion. The order matters: $\forall\exists$ does not imply $\exists\forall$ .

Order of quantifiers matters

Statement

Solution

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