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.