Types of Properties

A binary relation is said to be defined on a set X as having an equivalence relation if and only if the relation is reflexive, symmetric, and transitive. The connection can’t be an equivalence relation if any one of the three conditions—reflexive, symmetric, or transitive—is not met. The equivalence relation separates the set into distinct equivalence classes. If two items of the set belong to the same equivalence class, then and only then can we say that those elements are equal. An equivalence relation is generally denoted by the symbol ‘~’.

Equivalence Class 

An equivalence class is a subset B of A that satisfies the following condition: (a, b) R if and only if both a and b are members of B; a and b cannot exist outside of B. [a] = x A: (a, x) R is the mathematical notation for an equivalence class of a. The same equivalence class contains all of A’s items that are equal in some way. To phrase it another way, all of the elements that belong to the same equivalence class are interchangeable with one another.

A partition is a non-empty set of disjoint subsets of a set A, where no element of A is in more than one subset of A, and where elements that belong to the same subset are connected. Set A is equivalent to the union of all of the subsets that are contained within the partition.

A set of all equivalence classes of an equivalence relation is referred to as a quotient set, and it is expressed by the formula A/R = [a]: a A. Quotient Sets

Properties of Equivalence Relation

An illustration of an equivalence relation is provided below as an illustration of the properties.

Let there be a relation called R on the set of ordered pairs of positive integers if and only if the condition that ((a, b), (c, d)) R holds true. 

To demonstrate that R is an equivalence relation, we need to demonstrate that it is both reflexive and symmetrical, as well as transitive.

The following is the proof that satisfies the requirement that was given:

Property of Reflexivity

The reflexive characteristic is true for each aA if and only if the condition (a, a) R.

For all possible pairs of positive integers,

((a, b),(a, b))∈ R.

We can unequivocally say

For all positive integers, ab Equals ab.

As a direct consequence of this, the reflexive quality has been validated.

Property of Symmetry

Since it is symmetrical,

We can write (b, a) R if (a, b) R holds true.

Regarding the existing conditions,

If ((a, b),(c, d)) is true, then ((a, b)) is true.

ad = bc and cb = da if ((a, b),(c, d)) R

due to the commutative nature of the action of multiplication

As a result, ((c, d),(a, b)) R

The symmetric quality has been demonstrated in this manner.

Property of Transitivity

The transitive property tells us that:

If the pair (a, b) R and the pair (b, c) R are both R, then the pair (a, c) must also be R.

Given a set of ordered pairs of positive numbers, the following statements are true:

R (a, b), R (c, d), R (e, f), and R (c, d), R (e, f)

subsequently, R. (a, b) (e, f)

Let’s say that ((a, b), (c, d), (e, f)) R and ((c, d), (e, f)) R are both true.

Following that, we arrive at the conclusion that ad = cb, and cf = de.

As a result of the previously indicated relationship, we know that a/b = c/d and c/d = e/f.

The conclusion that can be drawn from this is that af = be.

As a result, ((a, b),(e, f)) R.

It can therefore be concluded that the transitive property exists.

Significance of Equivalence Relations

The equivalence relation is widely regarded as one of the most fundamental ideas in the field of mathematics. This is because it possesses several characteristics that are both distinctive and interesting. For instance, this can be accomplished through the application of an equivalence relation.

R⊂V×V

We can separate V into several distinct subsets, which we will refer to as its equivalence classes or partitions. In addition, each component of a class serves as a unique representation of the class as a whole. This means that if we can demonstrate a mathematical theorem for a single component of a class, we have demonstrated the theorem for the class as a whole. Because of this, equivalence classes are especially helpful for mathematicians; specifically, they save a great deal of research.

Conclusion

Three sets of properties are followed by a relation to be equivalent. These sets of properties are referred to as reflectivity, symmetry, and transitivity. In case even a single of these three is yet to be satisfied, a relation will not be considered equivalent. These properties of the equivalent relation have their own features and help in giving a new form to an existing function. Using these, we can unite or divide sets in as many parts as we want.