Ordered pair

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2010 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=35380