A set can contain any kind of thing as an element. In this lesson, our sets will contain tuples, or ordered coordinates. For example, \((4, 2, 7)\) is a tuple with \(4\) in the first coordinate, \(2\) in the second coordinate, and \(7\) in the third coordinate.

Two tuples are the same if, and only if, all of their coordinates are the same. For example, \((4, 2, 7) = (4, 2, 7),\) but \((4, 2, 7) \neq (2, 4, 7),\) because the first coordinates are not the same.

Definition: The cross product of the sets \(A\) and \(B\) is the set of pairs \(\{(a, b) : a \in A \mbox{ and } b \in B\}\).

The cross product of \(A\) and \(B\) is written \(A \times B\).

Example: Define \(A\) and \(B\) as \[A = \{1, 2\}, B=\{1, 3, 4\}.\] The cross product of \(A\) and \(B\) is \[A \times B = \{(1,1), (1,3), (1,4), (2,1), (2,3), (2,4)\}.\] You can also compute the cross product of a set with itself. \[A \times A = \{(1, 1), (1, 2), (2, 1), (2, 2)\}\] The set \(A \times A\) can also be written \(A^2.\)

You can visualize the cross product on a table. This table shows \(A \times B.\)