Cartesian Product (Set Product)

The Cartesian product (also known as the Cartesian product) forms ordered pairs from the elements of two sets. If A and B are sets, then the set A × B contains every ordered pair (a,b) where a ∈ A and b ∈ B. The order always matters: (a,b) ≠ (b,a), except if the two sets are identical and the elements are equal.

Simple Example

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

|A × B| = |A| · |B| = 2 · 3 = 6. Each element of A is paired with every element of B.

Properties

• The Cartesian product is not commutative in general: A × B ≠ B × A.
• It is associative: (A × B) × C = A × (B × C).
• The cardinality multiplies: |A × B| = |A| · |B|.

Applications

Cartesian products are fundamental in coordinate geometry (points (x,y)), functions (domain × codomain), and databases (relations as subsets of Cartesian products).

• In geometry: points in the plane are elements of ℝ × ℝ.
• In logic: truth tables as {true, false} × {true, false}.
• In computer science: pairs of keys and values in dictionaries.

Everyday Example

If there are 2 ice cream flavors (chocolate, vanilla) and 3 toppings (nuts, coconut, chocolate sauce), then every choice is an ordered pair (flavor, topping). Total 2 × 3 = 6 combinations possible.

Summary

The Cartesian product forms a new set of ordered pairs or multi-element sequences from sets. It plays a fundamental role in geometry, logic, computer science, and combinatorics.

Practice Exercise