Cartesian Product

The Cartesian product contains all possible ordered pairs from two sets. If sets A and B are given, then A × B is defined as follows:

The first element always comes from set A, the second from set B. It is important that (a,b) and (b,a) are generally not the same.

Simple Example

Let A = {1,2} and B = {x,y,z}. Then all pairings of A × B are:

It is visible that every element in A is paired with every element in B.

Cartesian Product of Multiple Sets

The Cartesian product is not only defined for two sets. It can also be formed for three or more sets, resulting in ordered triples, quadruples, etc.

Connection to Relations

A relation is always a subset of the A × B Cartesian product. Therefore, to understand relations, the concept of Cartesian product must first be clarified.

Everyday Examples

• If A = {"red", "blue"} and B = {"car", "bicycle"}, then A × B contains all possible pairings between colors and vehicles.
• If A = people, B = sports, then A × B contains all (person, sport) pairs.
• If A = days, B = hours, then A × B gives all possible time slots for a full timetable.

Summary

• The Cartesian product contains all ordered pairs from two sets.
• (a,b) ≠ (b,a), except if a = b.
• Cartesian products of multiple sets can also be formed, in the form of ordered triples, quadruples, etc.
• The concept of relation can be interpreted as a subset of A × B.

Practice Exercise