# Contents

## Idea

In material set theory, sets and elements are the same thing, so unordered pairs and pair sets are the same thing. However, in other foundations of mathematics, sets and elements are not the same thing, so an unordered pair is an element, while a pair set is a set.

## Definition

### In material set theory

If $A$ is a set and $x$ and $y$ are elements of $A$, then the pair set $\{x,y\}$ is a subset of $A$ such that for all elements for which $z \in \{x,y\}$ holds, $z = x$ or $z = y$. In material set theory, these are also called unordered pairs.

In material set theory, we may apply this when $x$ and $y$ are not previously given as elements of any set $A$. In that case, the existence of the unordered pair is given by the axiom of pairing.

### In structural set theory

If $A$ is a set and $x$ and $y$ are elements of $A$, then the pair set $\{x,y\}$ is a subset of $A$ with injection $i:\{x,y\} \hookrightarrow A$ such that for all elements $z \in \{x,y\}$, $i(z) = x$ or $i(z) = y$, and for all other sets $B$ with injection $j:B \hookrightarrow A$ such that for all elements $w \in B$, $i(w) = x$ or $i(w) = y$, there is an injection $k:B \hookrightarrow \{x,y\}$ such that for all elements $w \in B$, $i(k(w)) = j(w)$.

## Properties

The pair set $\{x,x\} = \{x\}$ is a singleton.