# Definition:Set Equality

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## Definition

Let $S$ and $T$ be sets.

### Definition 1

$S$ and $T$ are equal if and only if they have the same elements:

- $S = T \iff \paren {\forall x: x \in S \iff x \in T}$

### Definition 2

$S$ and $T$ are equal if and only if both:

- $S$ is a subset of $T$

and

- $T$ is a subset of $S$

## Axiom of Extension

The concept of set equality is axiomatised as the Axiom of Extension in the axiom schemata of all formulations of axiomatic set theory:

Let $A$ and $B$ be sets.

The **axiom of extension** states that:

- $A$ and $B$ are equal

- they contain the same elements.

That is, if and only if:

and:

This can be formulated as follows:

- $\forall x: \paren {x \in A \iff x \in B} \iff A = B$

## Equality as applied to Classes

In the context of class theory, the same definition applies:

Let $A$ and $B$ be classes.

### Definition 1

$A$ and $B$ are **equal**, denoted $A = B$, if and only if:

- $\forall x: \paren {x \in A \iff x \in B}$

where $\in$ denotes class membership.

### Definition 2

$A$ and $B$ are **equal**, denoted $A = B$, if and only if:

- $A \subseteq B$ and $B \subseteq A$

where $\subseteq$ denotes the subclass relation.