"Truth table" for quantifiers

"For every x (in the domain), there is at least one y (in the domain) that can depend on x and may be equal to x such that L(x, y) is true"
- $$\forall$$ x $$\exists$$ y L(x, y)
- Everyone loves someone (e.g. their family)

"There is at least one y such that for every x (including the case y=x), L(x, y) is true"
- $$\exists$$ y $$\forall$$ x L(x, y) 
- There is someone everyone loves (e.g. santa)

Nested Quantifiers

De Morgan's law (kind of) but extended...

$$\lnot$$ $$\forall$$ x P(x) $$\equiv$$ $$\exists$$ x $$\lnot$$ P(x)
- There is an x for which P(x) is false

$$\lnot$$ $$\exists$$ x  P(x) $$\equiv$$ $$\forall$$ x $$\lnot$$ P(x)
- For every x, P(x) is false

Negating quantifiers

Proof with quantifiers

 $$\exists$$ : "there exists", "at least one", "some"

"someone will buy an umbrella"
$$\exists$$ x B(x, umbrella)

University of Michigan - Ann Arbor

How to refer to someone or everyone (at least one or all)...

Universal quantifier ($$\forall$$)
- "for all", "every", "any"

Existential quantifier ($$\exists$$)
- "there exists", "at least one", "some"

Predicates and quantification

University of Michigan EECS 203 Lecture Notes

$$\forall$$: "for all", "every", "any"
- all includes the universe of discourse or domain of discourse

"everyone will buy an umbrella"
- $$\forall$$ x B(x, umbrella)

Universal quantifier

Existential qualifier

Two statements with quantifiers and predicates are logically equivalent if they have the same truth table regardless of the meaning of the predicates and the domain of discourse

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