Exercise 1046Logic, sets and proofs › Proof techniquesL1Added 3 Nov 2026

Order of quantifiers

#quantifiers#order#logic

Statement

Explain the difference between:
- $P_1$ : $\forall x\in\mathbb{R}$ , $\exists y\in\mathbb{R}$ , $y>x$ ;
- $P_2$ : $\exists y\in\mathbb{R}$ , $\forall x\in\mathbb{R}$ , $y>x$ .

Hint

$P_1$ allows $y$ to depend on $x$ ; $P_2$ requires one $y$ for all $x$ .

Show solutionHide solution

Solution

$P_1$ is true. For every real $x$ , choose $y=x+1>x$ .

$P_2$ is false. No single real number is greater than every real.

Conclusion. Order of quantifiers matters: $\forall\exists$ is weaker than $\exists\forall$ .

Order of quantifiers

Statement

Solution

More problems in Proof techniques