# Definition:Quantifier

## Contents

## 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 known as

Some sources 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 $x + y = 0$.*

## 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 - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{III}$: The Logic of Predicates $(1): \ 3$: Quantifiers - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts