Equivalence Classes and Cardinality

Equivalence classes are sets that contain elements that are equivalent to each other based on a certain relation. For example, if we consider the relation of congruence modulo 5, then the equivalence class of 2 would contain all integers that leave a remainder of 2 when divided by 5, such as {2, 7, 12, -3, ...}.

Cardinality is a measure of the size of a set, which is given by the number of elements in the set. For example, the cardinality of the set {1, 2, 3, 4, 5} is 5.

Equivalence Classes of a Set A

If we consider the set A and a certain relation R on A, then the equivalence classes of A under R form a partition of A. This means that every element of A belongs to exactly one equivalence class, and two elements belong to the same equivalence class if and only if they are equivalent under R.

For example, if we consider the set A = {1, 2, 3, 4, 5} and the relation R of congruence modulo 2, then the equivalence classes of A under R would be {1, 3, 5} and {2, 4}.

Cardinality of Equivalence Classes

The cardinality of an equivalence class is the number of elements in the equivalence class. If we denote the cardinality of an equivalence class [a] by |[a]|, then we have:

|[a]| = |{x ∈ A | xRa}|,

where a is an element of A, and Ra denotes the equivalence relation between a and x.

If we consider all the equivalence classes of A under R, then these equivalence classes will form a partition of A. This means that every element of A belongs to exactly one equivalence class, and the union of all the equivalence classes is equal to A.

Therefore, we have:

|A| = ∑ |[a]|,

where the sum is taken over all equivalence classes of A under R.

Since every element of A belongs to exactly one equivalence class, we can also write:

|A| = ∑ |[a]| = ∑ |[x]|,

where the sum is taken over all elements x of A.

This means that the set of all equivalence classes of A has the same cardinality as A.