Definition:Set Equality/Definition 2
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Definition
Let $S$ and $T$ be sets.
$S$ and $T$ are equal if and only if both:
- $S$ is a subset of $T$
and
- $T$ is a subset of $S$
Notation
This can be denoted in several ways:
- $S = T \iff \paren {S \subseteq T} \land \paren {T \subseteq S}$
or:
- $S = T \iff \paren {S \subseteq T} \land \paren {S \supseteq T}$
or:
- $S = T \iff S \subseteq T \subseteq S$
Equality of Classes
In the context of class theory, the same definition applies.
Let $A$ and $B$ be classes.
$A$ and $B$ are equal, denoted $A = B$, if and only if:
- $A \subseteq B$ and $B \subseteq A$
where $\subseteq$ denotes the subclass relation.
Also see
Sources
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