Definition:Set Equality

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Definition

Definition 1

Two sets are equal if and only if they have the same elements.


This can be defined rigorously as:

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

where $S$ and $T$ are both sets.


Definition 2

Let $S$ and $T$ be sets.

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

$S$ is a subset of $T$

and

$T$ is a subset of $S$


Axiomatic Set Theory

The concept of set equality is axiomatised in the Axiom of Extension in Zermelo-Fraenkel set theory:


Two sets are equal if and only if they contain the same elements:

$\forall x: \left({x \in A \iff x \in B}\right) \iff A = B$

The order of the elements in the sets is immaterial.


Also see