Difference between revisions of "Cartesian product"
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The '''Cartesian product''' of two [[set]]s <math>A</math> and <math>B</math> is the set of all [[ordered pair]]s <math>(a,b)</math> such that <math>a</math> is an [[element]] of <math>A</math> and <math>b</math> is an [[element]] of <math>B</math>. More generally, the Cartesian product of an ordered family of sets <math>A_1, A_2, \dotsc</math> is the set <math>A_1 \times A_2 \times \dotsb</math> of [[ordered tuples]] <math>(a_1, a_2, \dotsb)</math> such that <math>a_j</math> is an element of <math>A_j</math>, for any positive integer <math>j</math> for which we have specified a set <math>A_j</math>. | The '''Cartesian product''' of two [[set]]s <math>A</math> and <math>B</math> is the set of all [[ordered pair]]s <math>(a,b)</math> such that <math>a</math> is an [[element]] of <math>A</math> and <math>b</math> is an [[element]] of <math>B</math>. More generally, the Cartesian product of an ordered family of sets <math>A_1, A_2, \dotsc</math> is the set <math>A_1 \times A_2 \times \dotsb</math> of [[ordered tuples]] <math>(a_1, a_2, \dotsb)</math> such that <math>a_j</math> is an element of <math>A_j</math>, for any positive integer <math>j</math> for which we have specified a set <math>A_j</math>. | ||
| − | == | + | == 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>. | ||
== See Also == | == See Also == | ||
Revision as of 17:11, 29 August 2024
The Cartesian product of two sets 
and 
is the set of all ordered pairs 
such that 
is an element of and 
is an element of . More generally, the Cartesian product of an ordered family of sets 
is the set 
of ordered tuples 
such that 
is an element of 
, for any positive integer 
for which we have specified a set .
Existence
Ordered Pairs
In the language of set theory, it is not trivial to define an ordered pair since the set 
and 
are equivalent. Thus, the definition of an ordered pair is the set 
Through this definition, the pair does not equal the pair 
since the set and 
are not equivalent. However, for the ordered pair 
the resulting set reduces to 
(do you see why?). Thus reversing the positions of in the ordered pair does not change the resulting set.
Generally, the ordered pair 
can be though of as nested ordered pairs: 
.
See Also
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