Types of Relations & Equivalent Relations

If and only if a relation R on a set A is reflexive, symmetric, and transitive, it is said to be an equivalence relation. The equivalence relation is a set relationship that is usually denoted by the symbol “∼”.An equivalence relation defined on a set is a binary relation that is reflexive, symmetric, and transitive in mathematics. A binary relation over the sets A and B consisting of components of the type (a, b) such that an A and b B is a subset of the Cartesian product A B. The reflexive, symmetric, and transitive equivalence relation ‘equal to (=)’ is a common and simple example of an equivalence relation. As the names suggest, two components of a set are said to be equal if and only if they belong to the same equivalence class.

Equivalent Relations

A binary equivalence relation is one that 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 equivalence classes that are distinct. 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.

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 together. It meets the following requirements for all elements a, b, and c ∈ A:

• 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

Equivalence Relation Examples

• ‘Is equal to (=)’ is an equivalence relation on any set of integers A, as we have a = a, a = b, b = a, and a = b, b = a for all components a, b, c ∈ A.
• ⇒ a = c is a formula that can be used to calculate the value of a variable. This means (=) is transitive, symmetric, and reflexive.
• ‘Is reflexive, symmetric, and transitive, like (~) defined on the set of triangles?
• ‘Has the same birthday’ is reflexive, symmetrical, and transitive when applied to a group of persons.
• The equivalence relation ‘Is congruent to’ defined on the set of triangles is reflexive, symmetric, and transitive.
• On the set of integers, ‘congruence modulo n (≡)’ is defined as follows: It’s transitive, symmetric, and reflexive.
• The equivalence relation ‘has the same absolute value’ defined on the set of real numbers is reflexive, symmetric, and transitive.

Types of Relation in math

A subset of the Cartesian product of A and B, which is denoted by AXB, is a relation from set A to set B. The attributes of relations are the sorts of relations. There are several sorts of relations, including reflexive, symmetric, transitive, and anti-symmetric, which are described and explained using real-world examples.

Several forms of relations in math, for example

1. Reflexive
2. symmetric, 
3. Transitive
4. Anti-symmetric
5. Empty Relation
6. Equivalence relation

Reflexive relation: If (a, a) € R for every a € R, a relation R is said to be reflexive over a set A.

Eg: The relation “is brother of” is not reflexive over A if A is the set of all males in a family. Because no member of set A can be a brother of another member of the set.

Symmetric: A symmetric relation is one in which both parties are equal. If (a, b) € R => (b, a) € R, R is said to be symmetric.

Eg: The relation “is brother of” is symmetric over A if A is the set of all males in a family. Because if an is b’s brother, then b is a’s brother.

Transitive:  If (a, b) € R, (b, c) € R => (a, c) € R, the relation R is said to be Transitive.

Eg: The relation “is brother of” is transitive over A if A is the set of all males in a family. Because if a has a brother named b and b has a brother named c, then a has a brother named c.

Anti-symmetric: 

In a formal sense, relation R is antisymmetric if, for all a and b in A, if R(x, y) with x y, then R(y, x) must not hold, or if R(x, y) and R(y, x), then x = y.

Empty Relation:

The relation is known as an Empty Relation if no element of set X is related or plotted to any of the elements of set Y.

Equivalence relation: 

If (and only if) a relation is Transitive, Symmetric, and Reflexive, it is said to be an equivalence relation.

Conclusion: 

An equivalence relation is a binary relation that is reflexive, symmetric, and transitive in mathematics. The connection is equivalent to is the basic example of an equivalence relation. Each equivalence relation divides the underlying set into separate equivalence classes. Two elements of a given set are equivalent if and only if they belong to the same equivalence class. An equivalence relation defined on a set is a binary relation that is reflexive, symmetric, and transitive in mathematics. A binary relation over the sets A and B consisting of components of the form (a, b) such that an A and b B are a subset of the cartesian product A B.

Reflexivity, symmetry, and transitivity are always immediate for a certain equivalence relation (at least from what I have seen). You can use equivalence relations to define modulo, for example. But let’s pretend you’ve never heard of equivalence relations. Modulo could still be defined without issue.