Q.Let $A = \{a, b, c\}$, $B = \{1, 2\}$. Find $A \times B$ and $B \times A$. Is $A \times B = B \times A$?

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"?

The name comes from René Descartes, who invented the coordinate plane. The $xy$-plane you know is actually $\mathbb{R} \times \mathbb{R}$ — every point $(x, y)$ is an ordered pair where $x$ and $y$ are real numbers. That's the most famous Cartesian product of all.

Key points to remember

• Each element of $A \times B$ is an ordered pair — the first component comes from $A$, the second from $B$.
• If either $A$ or $B$ is empty, then $A \times B = \varnothing$. You can't form a pair if one set has nothing to contribute.
• The product can be extended to more than two sets: $A \times B \times C$ is the set of ordered triples $(a, b, c)$.