Examples of Relations

Relations are not only abstract mathematical concepts but appear in many everyday and scientific areas. Let's look at some simple examples.

Relations Between Numbers

• "x < y" → the less than relation: if x is less than y, then (x,y) is in the relation.
• "x = y" → the equality relation: every number is related to itself.
• "x is divisible by y" → if y divides x, then (x,y) is in the relation.

Relations Between People

• "x is sibling of y" → symmetric, but not transitive.
• "x is parent of y" → not symmetric and not transitive (composition gives 'grandparent').
• "x is friend of y" → usually symmetric, but not always transitive.

Everyday Examples

• "x goes to the same class as y" → equivalence relation: reflexive, symmetric, transitive.
• "x is ahead in line before y" → strict ordering: transitive and asymmetric.
• "x was born on the same day as y" → equivalence relation.

Mathematical Relations in Practice

In mathematics, relations are used, for example, to define graphs, determine functions, or connect tables in databases. Everywhere, the basic principle is the same: connection between elements of two sets.

Summary

Relations appear in every area of life: between numbers, people, and everyday situations. Among them are equivalences, orderings, and many other types, which we will get to know in detail later.

Practice Exercise