Start Learning Sets 4 | Cartesian Product and Maps [dark version]

Cartesian product and maps are built from the same core idea: pairing elements from two sets, then using those pairs to encode a rule.

The Cartesian product of two sets A and B—written A × B—is the set of all ordered pairs (a, b) where the first entry a comes from A and the second entry b comes from B. Order matters: swapping the entries changes the pair, so (a, b) is not the same as (b, a) unless the sets and elements line up in a way that makes the ordered pairs identical. A concrete example uses A = {triangle, square, circle} and B = {4, 7}. The Cartesian product contains six ordered pairs: (triangle, 4), (triangle, 7), (square, 4), (square, 7), (circle, 4), and (circle, 7). For intuition, the transcript suggests visualizing A as an x-axis and B as a y-axis, turning A × B into a grid of coordinate positions—each position corresponds to exactly one ordered pair.

To connect this “ordered” object back to ordinary set theory, ordered pairs are defined using sets. One common encoding is: (X, Y) is represented by a set whose elements are {X} and {X, Y}. With this encoding, equality of ordered pairs reduces to equality of the underlying components: (X, Y) = (X̃, Ỹ) happens exactly when X = X̃ and Y = Ỹ. Once ordered pairs are secured this way, the Cartesian product can be treated purely as a set of these encoded ordered pairs.

The next step turns a subset of A × B into a map (also called a function). A subset Gf ⊆ A × B is called the graph of a map f if it satisfies the “uniqueness” requirement: for every X in A, there is exactly one Y in B such that (X, Y) lies in Gf. In the plane picture, that means each x-value from A can correspond to at most one y-value in B; the graph should behave like a one-dimensional rule rather than a scatter of multiple points for the same input. The transcript also emphasizes the “exactly one” part: every element of the domain must be assigned an output.

This leads to standard notation. A map is written f: A → B, and the statement “f(x) = y” means that the ordered pair (x, y) belongs to the graph Gf. Here, A is the domain (the set of inputs) and B is the codomain (the set of potential outputs). An example uses the same A as before and a numerical set B. The map can send triangle to 1 and circle to 6, while the square can be sent to either 1 or 6 (or another allowed number if present). The key asymmetry remains: every element in A must be hit exactly once as an input, but elements in B may be unused or hit multiple times as outputs.

Overall, the framework is systematic: Cartesian products create the universe of ordered pairs, and maps are precisely the subsets of that universe that assign each domain element a unique codomain value.