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Cartesian Product Definition, Formula & Examples - Lesson

In the previous section, the Cartesian product of sets for a generic number of sets and for two sets as well. In both cases, some elements were written down, but suspension points were used to indicate that it is expected that the reader knows how to complete the missing elements. In this section, the Cartesian product of two sets with generic few elements is going to be provided. Consider two sets {eq}A~=~\{a_1,~a_2,~a_3\} {/eq} and {eq}B~=~\{b_1,~b_2,~b_3,~b_4\} {/eq}. The Cartesian product {eq}A~\times~B {/eq}, as seen previously, is the set of all ordered pairs, meaning that the first element must be in {eq}A {/eq}, and the second in {eq}B {/eq}. One practical way of making sure that no elements will be left out is to fix one element from the first set and vary the elements of the second one. Fixing {eq}a_1 {/eq} and varying the elements from {eq}B {/eq} gives the ordered pairs {eq}(a_1,~b_1),~(a_1,~b_2),~(a_1,~b_3),~(a_1,~b_4) {/eq}. Repeating the procedure for the other elements in {eq}A {/eq} gives the Cartesian product model {eq}A~\times~B~=~\{(a_1,~b_1),~(a_1,~b_2),~(a_1,~b_3),~(a_1,~b_4),~(a_2,~b_1),~(a_2,~b_2),~(a_2,~b_3),~(a_2,~b_4),~(a_3,~b_1),~(a_3,~b_2),~(a_3,~b_3),~(a_3,~b_4)\} {/eq}

Keep in mind that the Cartesian product is a set, so the order of the elements does not matter. But, because each element is an ordered pair, the order within the element does matter. For example, element {eq}(b_1,~a_1) {/eq} is not an element of {eq}A~\times~B {/eq}, but elements {eq}(a_1,~b_1),~(a_2,b_2) {/eq} can show up in any position in the Cartesian product of {eq}A {/eq} and {eq}B {/eq}.

In this section, a couple of examples are going to be shown to justify why the elements of a Cartesian product of two sets consist of ordered pairs. Consider sets {eq}A~=~\{1,~2,~3\} {/eq} and {eq}B~=~\{4,~5\} {/eq}. Determine {eq}A~\times~B {/eq} and {eq}B~\times~A {/eq}.

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