Quantifier
Definition
A quantifier is a logical operator that specifies the quantity of elements in a domain that satisfy a given predicate.
Types
Universal Quantifier (\(\forall\))
\[\forall x \in S, P(x)\]
"For all \(x\) in \(S\), property \(P\) holds."
Existential Quantifier (\(\exists\))
\[\exists x \in S, P(x)\]
"There exists an \(x\) in \(S\) such that property \(P\) holds."
Uniqueness Quantifier (\(\exists!\))
\[\exists! x \in S, P(x)\]
"There exists exactly one \(x\) in \(S\) such that \(P\) holds."
Negation
| Original | Negation |
|---|---|
| \(\forall x, P(x)\) | \(\exists x, \neg P(x)\) |
| \(\exists x, P(x)\) | \(\forall x, \neg P(x)\) |
Examples
Example 1
\[\forall n \in \mathbb{N}, n \geq 0\]
"Every natural number is non-negative."
Example 2
\[\exists n \in \mathbb{N}, n \text{ is even}\]
"There exists an even natural number." (True: \(n = 2\))
Related
- Proposition — Simple statements without quantifiers
- Set — The domain of quantification
- Truth Table — Quantifiers extend propositional logic