4 Equivalence relations
4.1 What is a relation?
In this final section we look at a method of classifying the elements of a set by sorting them into subsets. We shall require that the set is sorted into disjoint subsets – so each element of the set belongs to exactly one subset. Such a classification is known as a partition of a set. In order to achieve a partition, we need to have a method which enables us to decide whether or not one element belongs to the same subset as another. We look first at the general idea of a relation, and then at the particular properties needed by a relation in order to partition a set. A relation which satisfies these special properties is known as an equivalence relation, and the subsets into which the set is partitioned are called equivalence classes.
Equivalence relations occur in all branches of mathematics. For example, in geometry, two possible relations between the set of all triangles in the plane are is congruent to and is similar to. These are both equivalence relations: the relation is congruent to partitions the set of triangles into classes such that all the triangles within each class are congruent to each other, whereas is similar to partitions the set into classes of similar triangles. These partitions are different: triangles of the same shape but different sizes are similar to, but not congruent to, each other.
Equivalence relations are not confined to sets of mathematical objects. For example, relations between people such as is the same height as and has the same birthday as are equivalence relations.
We shall use the symbol (known as tilde or twiddle) to represent a relation between two elements of a set.
Some texts use ρ, rather than , for an arbitrary relation. Certain relations have special symbols; for example,
- < means is less than,
- = means is equal to.
We say that is a relation on a set X if, whenever x, y X, the statement x y is either true or false.
- If x y is true, then x is related to y.
- If x y is false, then x is not related to y and we write x y.
The statement x y can be read as ‘x is related to y’ or ‘x twiddles y’.
- 1. The condition ‘is equal to’ is a relation on any set of real numbers because, for any x, y in the set, the statement ‘x is equal to y’ is either definitely true or definitely false. This relation is usually denoted by the symbol =. For this relation, each real number in the set is related only to itself!
- 2. The condition ‘is less than’ is a relation on any set of real numbers, and we usually denote it by the symbol <. For example, −2 < 1, but 1 −2 and 3 3.
- 3. The condition ‘is the derivative of’ is a relation on any set of functions. We can define
- For example, let f(x) = x3, g(x) = 3x2 and h(x) = 2ex. Then g f because g is the derivative of f, and h h because h is the derivative of h, but f g because f is not the derivative of g.
- 4. On , we can define a relation
- that is, z1 is related to z2 if the distance between z1 and z2 in the complex plane is less than or equal to 4. For example, 1 + i 2 − i because
- but 1 + i 3+ 5i because