# 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.