Introduction to Relations

Relation is one of the basic concepts in mathematics. Its essence is to create a connection between the elements of two sets. If there is an A and a B set, then the relation is nothing more than a subset of the A × B Cartesian product.

This means that every relation is a set of ordered pairs in which the first element comes from A and the second element from B.

Simple Example

Let A = {1,2,3} and B = {x,y}. Then all pairings of A × B: {(1,x), (1,y), (2,x), (2,y), (3,x), (3,y)}.

From these, any subset can be a relation. For example, R = {(1,x), (3,y)} is a valid relation.

Everyday Examples

• "x is friend of y" → connection between people.
• "x is smaller than y" → connection between numbers.
• "x goes to the same class as y" → equivalence-like relation.

Why Are Relations Important?

Relations allow us to describe and examine connections in general. They are indispensable in logic, set theory, graph theory, databases, and computer science.

Summary

A relation is thus nothing more than a connection between elements of two sets, described using ordered pairs. This seemingly simple concept forms the basis for many further mathematical structures and concepts.

Practice Exercise