Equivalence Relations

Equivalence Relation Definition

A binary relation is reflexive, symmetric, and transitive and is defined on a set X. The relation cannot be an equivalence relation if any of the three conditions—reflexive, symmetric, and transitive—are not met. The equivalence relation separates the set into distinct equivalence classes. If and only if two elements of the set belong to the same equivalence class, they are said to be equivalent. The sign ~ is commonly used to represent an equivalence relation.

Equivalence Relations Formal Definition

Mathematical relations for real numbers if and only if R defined on a set A is reflexive, symmetric, and transitive, it is said to be an equivalence relation. They’re frequently used to group comparable or equivalent objects. It meets the following requirements for all elements a, b, and c ∈ A:

Following are the satisfied conditions:

Reflexive – If (a, a) ∈ R for all a ∈ A, then R is reflexive.

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

Equivalence relations can be explained in the following terms

• The ‘is equal to (=)’ sign is used on a set of numbers; 

• For a set of triangles that is given, the relation of ‘is similar to (~)’ and ‘is congruent to (≅)’ shows equivalence.

•  The relation of ‘congruence modulo n (≡)’ shows equivalence for a set of integers given.

• Equivalence relations have the same cosine refers to a set of all angles. 

• Equivalence relations have the same absolute value for the collection of all real numbers. 

Other relationships:

• A partial order is all, transitive, reflexive, and antisymmetric relation.

• Equality is both a partial order and an equivalence relationship. On a set, equality is the only reflexive, symmetrical, and antisymmetric relation. In algebraic expressions, equal variables can be substituted for one another. Persons within the equivalence classes of an equivalence relation can substitute for each other, but not individuals within a class.

• A preorder is both transitive and reflexive in nature.

• A reflexive and symmetric connection is a dependency relation (if finite) or a tolerance relation (if infinite).

• The ternary equivalence relation is the ternary equivalent of the conventional (binary) equivalence relation.

Proving equivalence relation

You must show reflexivity, symmetry, and transitivity to prove an equivalence relation, therefore in our case, we can say:

Let’s take this as an example:

R in the set of integers is defined as R = {(a,b) | a + b is an integer} 

• Reflexivity: Because a + a = 2a and 2a is an integer, (a, a)∈ R, proving R is reflexive.

• Transitivity: If an integer is a + b and b + c, then consequently R is transitive as (a + b) + (b + c) = a +2b + c is also an integer

• Symmetry: If a + b is an integer, then b + a is also an integer. This demonstrates that if (a, b) ∈ R, then (b, a)∈ R; thus, R is symmetric.

As a result, we have demonstrated that R is an equivalence relation because it is reflexive, symmetric, and transitive.

Equivalence Class

Equivalence classes are also a distinct subset of equivalence relations.

Let R be a set A equivalence relation. The equivalence class of an is the set of all items that are connected to an element an of A. To put it another way, if R is an equivalence relation on A, the element a’s equivalence class is:

[a]R = {x  ∈ A | x ~ a} 

Conclusion

An equivalence relation is a type of binary relation that should be reflexive, symmetric, and transitive. The “equal to (=)” relation is a well-known example of an equivalence relation. To put it another way, we can determine whether two items of the provided set are equal if they belong to the same equivalence class. In mathematics, a relation is a link between two different collections of data. If two sets are considered, the relationship between them will be confirmed if the components of the two sets are linked.