Definition of Relations

In mathematical terms, a relation is a connection between two sets. Let A and B be two sets. By definition, a relation is a subset of the Cartesian product A × B.

This means that the relation consists of ordered pairs where the first element comes from A and the second from B.

Ordered Pairs

For an ordered pair (a,b), the first element always belongs to A, the second to B. This distinguishes the ordered pair from a set: (a,b) ≠ (b,a), except if a = b.

Binary Relation

If A = B, then the relation is called a binary relation. In this case, the relation is a subset of the Cartesian product A × A. Examples: less than or equal to, equal, divisibility.

General n-ary Relations

Relations can also be defined for more than two sets, called n-ary relations. For three sets A, B, C, the relation is a subset of A × B × C.

Example: A ternary relation could describe coordinates in 3D space: (x,y,z) where x ∈ A, y ∈ B, z ∈ C.

Important Notes

Important: a relation itself can be any subset. There is no rule for which pairs must be in it. Strict rules only appear when examining special relations (e.g., reflexive, symmetric, transitive).

Example

Let A = {1,2,3} and B = {x,y}. All possible pairs of A × B: {(1,x), (1,y), (2,x), (2,y), (3,x), (3,y)}. One possible relation from these: R = {(1,x), (3,y)}.

This is a valid relation because the ordered pairs satisfy the condition that the first element is from A, the second from B.

Summary

• Relation: a subset of the Cartesian product A × B.
• The elements are ordered pairs: (a,b), where a ∈ A, b ∈ B.
• If A = B, we speak of a binary relation.
• A relation can be any subset; special properties are defined later.

Practice Exercise