# 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

## Proof of uniqueness

Before we may write $(a,b)$ we must make sure this is not ambiguous.

Proof that $(a,b)=(a',b')\iff[a=a'\wedge b=b']$

$\impliedby$

Clearly if $a=a'$ and $b=b'$ then $(a,b)=\{\{a\},\{a,b\}\}=\{\{a'\},\{a',b'\}\}=(a',b')$ and we're done.

$\implies$

Assume $(a,b)=\{\{a\},\{a,b\}\}=\{\{a'\},\{a',b'\}\}=(a',b')$.

If $a\ne b$ then we must have $\{a\}=\{a'\}$ and $\{a,b\}=\{a',b'\}$ (as clearly $\{a\}=\{a',b'\}$ is false, there are either 2 or 1 elements not contained in $\{a\}$ that are in $\{a',b'\}$ - namely $a'$ and $b'$)

Clearly $a=a'$, then $\{a,b\}=\{a',b'\}\implies b=b'$.

If $a=b$ then $(a,a)=\{\{a\},\{a,a\}\}=\{\{a\}\}$, we know $\{\{a\}\}=\{\{a'\},\{a',b'\}\}$ so again using the Set theory axioms (namely Extensionality) we see $a=a'=b'$ so $a=a'$ and $b=b'$ holds here too. This completes the proof.