Axiom of Extensionality

Classes and Sets are constructed from axioms. In this section we look at the Axiom of Extensionality.

We first recognise that if ${A}$ and ${B}$ are classes then to be equal means every class ${X}$ that has ${A}$ in it also has ${B}$ in it. In this description we immediately require a concept of membership. Set Theory is based on a fundamental membership operator ${\in}$ .

Definition: ${A=B \iff \forall X [A \in X \iff B \in X]}$

Axiom: ${\text{Axiom of Extensionality}}$
${A=B \implies \forall A \forall B\; [\forall x (x \in A \iff x \in B)]}$

If an element is in ${A}$ then it is in ${B}$ and if it is in ${B}$ then it is in ${A}$ . Extensionality means that the set is completely described by its members. There is no hidden essence beyond its elements.

Definition: ${\text{Let } A \text{ and } B \text{ be classes, }}$ ${A\subseteq B \iff (x \in A \implies x \in B)}$

If ${A \subseteq B}$ then we say ${A}$ is a subset of ${B}$ meaning that if an element is in ${A}$ then it is in B but if it is in ${B}$ it is not necessarily in ${A}$ .

Using the Axiom of Extensionality and the ${\in}$ operator we can develop core relationships.

Theorem: ${A=A}$

Proof:
${A=A \iff x \in A \; \land \; x \in A }$ $\Box$

Theorem: ${A=B \iff B=A}$

Proof:
${A=B \implies (x \in A \implies x \in B \; \land \;x \in B \implies x \in A)}$
${B = A \implies (x \in B \implies x \in A \; \land \; x \in A \implies x \in B)}$ $\Box$

Theorem: ${A=B \; \land \; B = C \implies A = C}$

Proof:
${A=B \implies (x \in A \implies x \in B \; \land \; x \in B \implies x \in A)}$
${B = C \implies (x \in B \implies x \in C \; \land \; x \in C \implies x \in B)}$
${\implies (x\in A \implies x\in C \; \land \; x \in C \implies x \in B \; \land \; (x \in B \implies x \in A))}$
${\implies (x\in A \implies x \in C \; \land \; x \in C \implies x \in A)}$ $\Box$

Theorem: ${A \subseteq B \land B \subseteq A \implies A = B}$

Proof:
${A \subseteq B \iff (x\in A \implies x \in B)}$
${B \subseteq A \iff (x\in B \implies x \in A)}$
${A \subseteq B \land B \subseteq A \iff (x\in A \implies x \in B) \; \land \; (x\in B \implies x \in A)}$
${\iff A=B}$ $\Box$

These relationships are useful. They offer proof, using the Axiom of Extensionality to relationships that are generally taken as self evident. They do not form new classes or sets. To build a set we need the Axiom of Schema of Separation. This a subject for another section.