Ordered pair

Kuratowski definition

An ordered pair $$(a,b)=\{\{a\},\{a,b\}\}$$ , this way $$(a,b)\ne(b,a)$$ .

Ordered pairs are vital in the study of relations which leads to functions

Proof of existence

It is easy to prove ordered pairs exist
Suppose we are given $$a,b$$ (so we can be sure they exist).

By the axiom of a pair we may create $$\{a,b\}$$ and $$\{a,a\}=\{a\}$$ , then we simply have a pair of these, thus $$\{\{a\},\{a,b\}\}$$ exists.

The axioms may be found here

[Expand]Set Theory

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