Difference between revisions of "Cartesian product"

Latest revision as of 17:14, 29 August 2024

The Cartesian product of two sets $A$ and $B$ is the set of all ordered pairs $(a,b)$ such that $a$ is an element of $A$ and $b$ is an element of $B$ . More generally, the Cartesian product of an ordered family of sets $A_1, A_2, \dotsc$ is the set $A_1 \times A_2 \times \dotsb$ of ordered tuples $(a_1, a_2, \dotsb)$ such that $a_j$ is an element of $A_j$ , for any positive integer $j$ for which we have specified a set $A_j$ .

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 == Existence ==
-=== Ordered Pairs ===
-In the language of set theory, it is not trivial to define an ordered pair since the set <math>\{a,b\}</math> and <math>\{b,a\}</math> are equivalent. Thus, the definition of an ordered pair <math>(a,b)</math> is the set <math>\{\{a\}, \{a,b\}\}</math> Through this definition, the pair <math>(a,b)</math> does not equal the pair <math>(b,a)</math> since the set <math>\{\{a\}, \{a,b\}\}</math> and <math>\{\{b\}, \{b,a\}\}</math> are not equivalent. However, for the ordered pair <math>(a,a)</math> the resulting set reduces to <math>{{a}}</math> (do you see why?). Thus reversing the positions of <math>a</math> in the ordered pair does not change the resulting set.
-Generally, the ordered pair <math>(a, b, c, \dots )</math> can be though of as nested ordered pairs: <math>(a, (b, (c, (\dots))))</math>.
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Difference between revisions of "Cartesian product"

Latest revision as of 17:14, 29 August 2024

Existence

See Also