Definition:Set Equality/Definition 1
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Definition
Let $S$ and $T$ be sets.
$S$ and $T$ are equal if and only if they have the same elements:
- $S = T \iff \paren {\forall x: x \in S \iff x \in T}$
Otherwise, $S$ and $T$ are distinct, or unequal.
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:
- $\forall x: \paren {x \in A \iff x \in B}$
where $\in$ denotes class membership.
Also see
Sources
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