# Ordered pair

From Encyclopedia of Mathematics

2020 Mathematics Subject Classification: *Primary:* 03E [MSN][ZBL]

A construct $(a,b)$ of two objects $a$ and $b$ in which order is significant; $(a,b)$ is not the same as $(b,a)$ unless $a=b$. Equality between ordered pairs is defined by $$ (a,b) = (c,d) \ \Leftrightarrow \ a=c \wedge b=d \ . $$

A realisation in terms of axiomatic set theory is to write $$ (a,b) = \{ \{a\} , \{a,b\} \} \ . $$

Given sets $A$ and $B$ the set of all ordered pairs $(a,b)$ with $a \in A$ and $b \in B$ is the Cartesian product $A \times B$.

Compare with unordered pair.

#### References

- P. R. Halmos,
*Naive Set Theory*, Undergraduate Texts in Mathematics, Springer (1960)**ISBN**0-387-90092-6

**How to Cite This Entry:**

Ordered pair.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Ordered_pair&oldid=54585