Cartesian Products

What is a Cartesian Product?

You’re ordering a meal. You pick one main and one drink.

• Mains: burger, pizza
• Drinks: cola, water

What are all possible meals you could order?

• (burger, cola)
• (burger, water)
• (pizza, cola)
• (pizza, water)

That’s the Cartesian product — every possible pairing.

The Notation

$A \times B$

This reads: “A cross B”

It means: pair each element of A with each element of B.

Step-by-Step Example

$A = \{1, 2\}$ $B = \{a, b, c\}$

Find $A \times B$:

Take each element of A and pair it with every element of B:

$A \times B = \{(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)\}$

Each element of A gets paired with every element of B.

Order Matters

The pairs are ordered pairs — first element from A, second from B.

$(1, a) \neq (a, 1)$

These are different pairs. The order tells you which set each element came from.

This also means:

$A \times B \neq B \times A$

Example:

$A = \{1\}, \quad B = \{a\}$

• $A \times B = \{(1, a)\}$
• $B \times A = \{(a, 1)\}$

Different results.

How Many Pairs?

If A has $m$ elements and B has $n$ elements:

$|A \times B| = m \times n$

That’s why it’s called the Cartesian product — you multiply the sizes.

Examples:

The Coordinate Plane

Here’s why this matters.

The x-y coordinate plane you’ve used since school? That’s a Cartesian product.

$\mathbb{R} \times \mathbb{R} = \mathbb{R}^2$

Every point on the plane is a pair (x, y) where:

• x comes from the real numbers (horizontal axis)
• y comes from the real numbers (vertical axis)

The point (3, 5) is just an element of $\mathbb{R} \times \mathbb{R}$.

$\mathbb{R}$ is the set of all real numbers.

Special Cases

Product with empty set:

$A \times \emptyset = \emptyset$

If there’s nothing to pair with, you get no pairs.

Product with itself:

$A \times A = A^2$

This is common. $\mathbb{R}^2$ means $\mathbb{R} \times \mathbb{R}$.

Example:

$A = \{0, 1\}$ $A^2 = \{(0, 0), (0, 1), (1, 0), (1, 1)\}$

The Formal Definition

$A \times B = \{(a, b) \mid a \in A \text{ and } b \in B\}$

This reads: “The set of all ordered pairs (a, b) where a is in A and b is in B.”

Cartesian product builds all possible pairings.