Question

Suppose \(R\) is an equivalence relation on a finite set \(A\), and every equivalence class has the same cardinality \(m\). Express \(|R|\) in terms of \(m\) and \(|A| .\)

Short answer

The cardinality of the equivalence relation R, denoted as |R|, would be \(|A| * m\).

Step-by-step solution

Understanding the definition of an equivalence class

An equivalence relation R on a set A partitions the set A into disjoint subsets called equivalence classes. In this case, it is also stated that each equivalence class has the same size, denoted by m.

Applying the given conditions

It is stated that all equivalence classes have the same size m. Also, because the equivalence classes partition the entire set, there are total \(|A| / m\) equivalence classes (since they all have the size m and combine to form the set A). Now we have to find out the size of R, the equivalence relation itself.

Calculating the cardinality of R

For each pair of elements in an equivalence class, there is one relationship. Therefore, each equivalence class contributes \(m^2\) pairs to the equivalence relation. So, if there are \(|A| / m\) equivalence classes each giving \(m^2\) pairs, the total size of the equivalence relation is \((|A| / m) * m^2 = |A| * m\).

Key concepts

Equivalence Class

When diving into the topic of equivalence classes, we approach a foundational concept in mathematics that deals with the categorization of elements within a set.
An equivalence class is a subset of a given set formed by an equivalence relation. An equivalence relation is a type of binaryrelation that is reflexive, symmetric, and transitive.

For instance, consider a simple set of integers and the equivalence relation of having the same remainder when divided by a number. This will group the integers into equivalence classes where each class contains numbers that give the same remainder—thus partitioning the set into mutually exclusive slices.

In the context of the exercise, the equivalence relation partitions set A into equivalence classes where each class has precisely m elements. This uniformity simplifies the process of determining the total relationships within the set.

Set Theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects.
It is a fundamental language of mathematics and forms the basis of many mathematical disciplines. A set is typically considered a well-defined collection of distinct objects, called elements or members of the set.

The power of set theory lies in its ability to encode and manipulate collections of objects using operations like union, intersection, and complement. It also allows us to talk about the size or cardinality of a set, which is a measure of the number of elements in the set, whether finite or infinite.

• In our exercise, we are dealing with a finite set, which is a set with a limited number of elements—as opposed to an infinite set which has no end.

Finite Set

A finite set is a set with a countable number of elements, where 'countable' means the number can reach an end.
The set of natural numbers less than 100, or the set of students in a classroom, are examples of finite sets as you can list all the members.

In contrast, an infinite set, like the set of all even numbers, may be listed but never finishes.

• In the case of our exercise, we are told that the set A is finite, which means we can perform operations like counting the number of equivalence classes and determining their total cardinality.

Cardinality

Cardinality refers to the number of elements in a set, serving as a notion of the set's size.
For finite sets, the cardinality is simply the count of distinct elements. For infinite sets, cardinality becomes a more complex concept that requires deeper theoretical concepts to understand.

When we discuss the cardinality of an equivalence relation on set A, as in our exercise, we're interested in the number of pairs of elements in A that are related by the equivalence relation.

• Understanding cardinality allows us to quantify and compare the sizes of sets and to make precise statements about the relationships within those sets, such as the number of relations within an equivalence relation.