Quantifier

Abstract

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$$\forall ,\exists$$

• Instead of manually assigning a specific value to a predicate variable to obtain a mathematical statement, quantifiers provide another way to specify how many values of a particular type can make the predicate a closed predicate

Implicit quantification

The Quantifiers are assumed without specified explicitly

Statements with Multiple Quantifiers

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• Predicate with more than one Quantifier

Order of Quantifiers

Unless the Quantifiers are of the same type. Otherwise, the meaning is different.

Given $P(x,y)$ is x loves y

1. $\forall x\exists y,P(x,y)$ in english: For all people x, there is a person y such that x loves y.

2. $\exists y\forall x,P(x,y)$ in english: There is a person y such that all people x, x loves y.

• The first one means for everyone (you, me, he), there is someone we love
• The second one means there is someone who is loved by everyone (you, me, he)