# Extensionality Axiom

The ax­iom of ex­ten­sion­al­ity is one of the fun­da­men­tal ax­ioms of set the­ory. Ba­si­cally, it pos­tu­lates the con­di­tion, by which two sets can be equal. This con­di­tion can be de­scribed as fol­lows: if any two sets have ex­actly the same mem­bers, then these sets are equal. A for­mal no­ta­tion of the ex­ten­sion­al­ity ax­iom can be writ­ten as:

$$\forall A \forall B : ( \forall x : (x \in A \iff x \in B) \Rightarrow A=B)$$

## Examples

• $$\{1,2\} = \{2,1\}$$, be­cause what­ever ob­ject we choose, it ei­ther be­longs to both of these sets ($$1$$ or $$2$$), or to nei­ther of them (e.g. $$5$$, $$73$$)

com­ment:

• If $$A = \{x \mid x = 2n \text{ for some integer } n \}$$ and $$B = \{x \mid x \text{ is even } \}$$, then $$A=B$$. The proof goes as fol­lows: $$\forall x : (x \in A \Leftrightarrow (x = 2n \text{ for some integer } n ) \Leftrightarrow (x/2 = n \text{ for some integer } n) \Leftrightarrow (x/2 \text{ is an integer}) \Leftrightarrow (x \text{ is even}) \Leftrightarrow x \in B)$$

that, if sim­plified, gives $$\forall x : (x \in A \iff x \in B)$$, which, by ex­ten­sion­al­ity, im­plies $$A=B$$

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Fix the for­mat­ting in the cur­rently commeneted ex­am­ple. Every new state­ment needs to be in a new line, lined up.