Cartesian Products

The cartesian product of two sets, $A$ and $B$ , is written $A \times B$ and consists of ordered pairs where the first element of the pair comes from the set $A$ and the second element of the pair comes from the set $B$ . Mathematically, we write

$A \times B = {(a, b) : a \in A and b \in B}$ .

You can think of cartesian products like an array where the rows are given by the elements in $A$ and the columns are given by the elements in $B$ . Then the elements in the cartesian product are given by pairs where the first member of the pair comes from the row and the second member of the pair comes from the column.

Example: If $A = {1, 2, 3}$ and $B = {1, a, b}$ .

Problem: Find $A \times B$ .

Solution: We can form a table as:

	1	a	b
1	(1, 1)	(1, a)	(1, b)
2	(2, 1)	(2, a)	(2, b)
3	(3, 1)	(3, a)	(3, b)

So we see that the cartesian product is the set:

$A \times B = {(1, 1), (1, a), (1, b), (2, 1), (2, a), (2, b), (3, 1), (3, a), (3, b)} .$ $◼$

Note: The order within the pairs matters. For instance, $(a, 1)$ is NOT an element in $A \times B$ . However, the order of the pairs within the curly brackets is irrelevant because elements in a set can be listed in any order.

Example: Let $U = {- 2, - 1, 0, 1, 2}$ and $S = {- 1, 0, 1}$ . Also let $A = {(x, y) : x \in S, y = x^{2}}$ and $B = {(x, y) : x \in U, y = x^{2}}$ .

Problem: Is it true that

(a) $A \subset B$ (b) $A \subset S \times S$ (c) $B \subset S \times S$ (d) $B \subset U \times U$

Solution: In the following video from Adult Student Resources at IUB, Anna solves this problem.

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