- fundamentals of set theory
- material set theory
- presentations of set theory
- structuralism in set theory
- class-set theory
- constructive set theory
- algebraic set theory

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.

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.

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 $w \in B$, $i(k(w)) = j(w)$.

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

Created on December 11, 2022 at 16:47:49. See the history of this page for a list of all contributions to it.