Q.If $A \times B = \{(a, x), (a, y), (b, x), (b, y)\}$, then find $A$ and $B$.

Cartesian Product of Two Sets

Imagine you have two trays of ingredients. One tray has bread options: white and brown. The other has fillings: cheese and jam. If you want to make a sandwich, you pick one bread and one filling. Every possible sandwich is a pair: (white, cheese), (white, jam), (brown, cheese), (brown, jam). That list of all possible ordered pairs — one from the first set, one from the second — is exactly the Cartesian product.

The idea is simple: take every element from the first set and pair it with every element from the second set. Order matters — (white, cheese) is not the same as (cheese, white), because the first slot always comes from the first set.

The precise definition

Let $A$ and $B$ be two sets. The Cartesian product of $A$ and $B$, written $A \times B$, is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

If $A$ has $m$ elements and $B$ has $n$ elements, then $A \times B$ has $m \times n$ elements. That's why it's called a product — the sizes multiply.

A concrete example

Let $A = \{1, 2\}$ and $B = \{x, y, z\}$.

Then $A \times B$ is:

Notice: 2 elements in $A$, 3 in $B$, and $2 \times 3 = 6$ ordered pairs in the product.

Why "Cartesian"? …