A General Introduction on Equivalence Relations

An equivalence relation in mathematics is a type of binary relation that needs to be transitive, symmetric, and reflexive. The “equal to (=)” relation is a well-known illustration of an equivalence relation. In other words, if two items of the given set are members of the same equivalence class, then they are equal to one another.

Equivalence Relations

A binary relation that is reflexive, symmetric, and transitive is called an equivalence relation when it is defined on a set in mathematics. Any element of the form (a, b) such that a A and b ϵB makes up a binary relation over the sets A and B. This relation is a subset of the cartesian product A B. The reflexive, symmetric, transitive “equal to (=)” relation is an example of an equivalence relation that is very common and simple to understand. Two components of a set are said to be equal, as the name implies, if and only if they fall under the same equivalence class.

A binary relation defined on a set X as an equivalence relation is reflexive, symmetric, and transitive. Any one of the three requirements—reflexive, symmetric, and transitive—must be met for the relation to qualify as an equivalence relation. As a result of the equivalence relation, the set is split into distinct equivalence classes. When and only when two elements of the set fall under the same equivalence class, they are deemed to be equal. The sign “∼” is typically used to indicate an equivalency connection.

Definition of Equivalence Relations

Mathematical relationships for real numbers If and only if R defined on a set A is reflexive, symmetric, and transitive, then R is said to be an equivalence relation. They are frequently used to put together items that are comparable or identical. For each of the elements a, b, and c∈A, it meets the requirements listed below:

• Reflexive – R is reflexive if (a, a) ∈ R for all a ∈ A
• Symmetric – R is symmetric if and only if (a, b) ∈ R ⇒ (b, a) ∈ R for all a, b ∈ A
• Transitive – R is transitive if and only if (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R for all a, b, c ∈ A

The reflexive relation, symmetric relation, and transitive relation are three different types of relations that make up the equivalence relation.

Proof of Equivalence Relations

Let’s look at an illustration. If and only if a = b, define a relation R on the set of natural numbers N as (a, b)∈ R. We shall now demonstrate the reflexivity, symmetry, and transitivity of the relation R.

For Reflexivity

Every natural number is equal to itself, which is known as the reflexive property i.e., a = a for all a ∈ N ⇒ (a, a) ∈ R for all a ∈ N. Hence, R is reflexive.

For Symmetric

For a, b ∈ N, let (a, b) ∈ R ⇒ a = b ⇒ b = a ⇒ (b, a) ∈ R. Since a, b are arbitrary, R is symmetric.

For Transitivity

For a, b, c ∈ N, let (a, b) ∈ R and (b, c) ∈ R ⇒ a = b and b = c ⇒ a = c (as numbers equal to the same number are equal to one another) ⇒ (a, c) ∈ R. Since a, b, c are arbitrary, R is transitive.

Reflexive, symmetric, and transitive properties characterise R as an equivalence relation on the set of natural numbers N.

Example of Equivalence Relations

Establish that the relation R is an equivalence relation for the set A ={1, 2, 3, 4,5} given by the relation R = (a, b):|a-b| is even.

Solution 

To prove given relation is equivalence relation we need to prove R is reflexive, symmetric and transitive.

For Reflexivity

Given the relationship,

|a – a| = | 0 |=0 Zero is also always even. Therefore, |a-a| is even. This means that (a, a) belongs to R. R is hence Reflexive.

For Symmetric

Given the relationship, |a – b| = |b – a|  We are aware that                                      |a – b| = |-(b – a)|=|b-a| Because |a – b| is equal, Consequently, |b – a| is also even. As a result, (b, a) belongs to R if (a, b)∈ R. R is hence symmetric.

For Transitivity

If |a-b| is even, (a-b) must also be even. In the same way, if (b-c) is even, then |b-c| must likewise be even. Even numbers make up the sum. It can therefore be written as a-b+ b-c is even. In addition, a – c is even. So,If |a-b| and |b-c| are equal, then |a-c| is also equal.

As a result, (a, c) also belongs to R if (a, b) ∈R and (b, c)ϵ R. R is hence transitive.

Conclusion

A binary connection that is reflexive, symmetric, and transitive is known as an equivalence relation in mathematics. One such illustration of an equivalence relation is the relation between line segments in geometry known as equipollence. A division of the underlying set into distinct equivalence classes is provided by each equivalence relation. Only if they are members of the same equivalence class are two items of the given set equal to one another.