Definition:Quantifier
Definition
The universal quantifier $\forall$ and the existential quantifier $\exists$ are referred to collectively as quantifiers.
The Universal Quantifier
The symbol $\forall$ is called the universal quantifier.
It expresses the fact that, in a particular universe of discourse, all objects have a particular property.
That is:
- $\forall x:$
means:
- For all objects $x$, it is true that ...
In the language of set theory, this can be formally defined:
- $\forall x \in S: \map P x := \set {x \in S: \map P x} = S$
where $S$ is some set and $\map P x$ is a propositional function on $S$.
The Existential Quantifier
The symbol $\exists$ is called the existential quantifier.
It expresses the fact that, in a particular universe of discourse, there exists (at least one) object having a particular property.
That is:
- $\exists x:$
means:
- There exists at least one object $x$ such that ...
In the language of set theory, this can be formally defined:
- $\exists x \in S: \map P x := \set {x \in S: \map P x} \ne \O$
where $S$ is some set and $\map P x$ is a propositional function on $S$.
Also defined as
Some sources explicitly categorize the quantifiers as operators.
Examples
Existence for All of Element Greater Than
- $\forall x: \exists y: x < y$
means:
- For every $x$ there exists a $y$ such that $x < y$
or (assuming the domain is that of numbers):
$\epsilon$-$\delta$ Condition
- $\forall \epsilon: \exists \delta: \forall y: \size {x - y} < \delta \implies \size {\map f x - \map f y} < \epsilon$
means:
- For every $\epsilon$ there exists a $\delta$ such that for every $y$:
- If $\size {x - y} < \delta$ then $\size {\map f x - \map f y} < \epsilon$.
Uniqueness of Additive Identity
- $\forall x: \exists ! y: x + y = 0$
means:
- For every $x$ there exists a unique $y$ such that $x + y = 0$.
Also see
- Results about quantifiers can be found here.
Sources
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S 1.4$: Universal and Existential Quantifiers
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Variables and quantifiers
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{III}$: The Logic of Predicates $(1): \ 3$: Quantifiers
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): quantifier
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): quantifier