 Ordered pair

An ordered pair $(a,b)$ records two elements together with their positions. Its defining equality rule is $(a,b)=(c,d)$ exactly when $a=c$ and $b=d$. Thus $(1,2)\ne (2,1)$, and repeated entries such as $(a,a)$ are allowed.

The Cartesian product $A\times B$ consists of the ordered pairs with first entry in $A$ and second entry in $B$. A function can be represented by a set of such pairs with exactly one output for each input. In set theory, a Kuratowski ordered pair realizes the equality rule using ordinary sets.

Sets are unordered, but we can use them to create ordered objects, which are of fundamental importance. Notably, they are used in the definition of functions.