Definition:Class Equality
Definition
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.
When $x$ is a set variable, equality of $x$ and $A$ is defined using the same formula:
- $x = A$ if and only if $\forall y: \paren {y \in x \iff y \in A}$
- $A = x$ if and only if $\forall y: \paren {y \in A \iff y \in x}$
Axiom of Extension
The concept of class equality is axiomatised as the Axiom of Extension:
Let $A$ and $B$ be classes.
Then:
- $\forall x: \paren {x \in A \iff x \in B} \iff A = B$
Equality as applied to Sets
In the context of set theory, the same definition applies:
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 $S$
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This definition "overloads" the $=$ sign, since $x = y$ could refer to either class equality or set equality.
However, this overloading is justified because for sets $x$ and $y$, $x = y$ is equal for either set equality or class equality.
This fact is proved on Class Equality Extension of Set Equality.
Also see
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 4.5$