Ordered pair

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