In mathematical terms a relation on a set S is simply a subset R ⊆ S × S. This definition is deceptively simple, but captures the real-world nature of relations remarkably. Of course, if one wants interesting mathematics one must restrict to relations with certain properties. The two most important types of relations in mathematics are equivalence relations and order relations.

Basic definitions and properties

Definition A.1.1 A relation R on a set S is a subset R ⊆ S × S. We will write xRy to mean (x, y) ∈ R.

Definition A.1.2 A relation R on S is reflexive if xRx for every x ∈ S, symmetric if xRy ⇒ yRx for every x, y ∈ S, antisymmetric if xRy ∧ yRx ⇒ x = y for every x, y ∈ S and transitive if xRy ∧ yRz ⇒ xRz for every x, y, z ∈ S.

1. (i) R is called an equivalence relation if it is reflexive, symmetric and transitive.

2. (ii) R is called a partial ordering if it is reflexive, antisymmetric and transitive.

Example A.1.3 Recall the relation ≤ on ℤ from Chapter 1 given by x ≤ y ⇔ y – x ∈ N. Since 0 ∈ N, ≤ is reflexive. It is antisymmetric since if x ∈ ℤ and x ∈ N, –x ∈ N then x = 0. It is transitive, as x, y ∈ N implies x + y ∈ N. So ≤ is a partial ordering on ℤ.