Equivalence Relations and Classes

A relation R on a set A is an equivalence relation if and only if it is reflexive, symmetric, and transitive.The equivalence relation is a set relationship that is usually denoted by the symbol “∼”.

• Reflexive: A relation is reflexive, if (a, a) ∈ R, for every a ∈ A.

• Symmetric: A relation symmetric, if (a, b) ∈ R, then (b, a) ∈ R.

• Transitive: A relation transitive if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

If and only if a binary relation on a set A is reflexive, symmetric, and transitive,then it is said to be an equivalence relation.

For all x, y, and z in set A, a relation R is said to be equivalence relation if:

• (x,x) R (Reflexivity)
• (x,y) R if only if (y,x) R (Symmetry)
• If (x,y) R and(y,z) R, then(x,z) R (Transitivity)

Definitions Related to Equivalence Relation

We shall now understand the meanings of various terms related to equivalence relations and classes, such as equivalence class, partition, quotient sets, and so forth. Consider the equivalence relation R, which is specified on the set A and has the following properties:a, b ∈ A.

Equivalence Class –

Mathematically, an equivalence class of a is denoted as [a] = {x ∈ A: (a, x) ∈ R}. This comprises all of A’s elements related to the letter ‘a’. The equivalence class for all items of A that are equivalent to one another is the same. To put it another way, all components in the same equivalence classes are equivalent to one another.

A partition of set A is a non-empty set of disjoint subsets of A in which no member of A appears in more than one subset and elements from the same subset are connected. Set A is equivalent to the union of the partition’s subsets.

A quotient set is a collection of all equivalence classes of an equivalence relation indicated by the symbol.

A/R = {[a]: a ∈ A}

Proof of Equivalence Relation

Let’s look at an example to see how to verify that a connection is an equivalence relation. Define R on the set of natural numbers N as follows: (a, b)∈ R if and only if a = b.  We’ll now demonstrate that R is reflexive, symmetric, and transitive

Reflexive Property-Since every natural number is identical to itself, it is known as the reflexive property, a = a for all a ∈ N ⇒ (a, a) ∈ R for all a ∈ N. As a result, R is reflexive.

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

Transitive Property – for all a,b,c ∈ N, let (a,b) ∈ R and (b,c) ∈ R ⇒ a = b and b = c⇒ a = c ⇒ (a,c) ∈ R. R is transitive because a, b, and c are arbitrary.

R is an equivalence relation since it is reflexive, symmetric, & transitive when specified on the set of natural numbers N.

Equivalence classes- The subset of S that includes all components equal to each other is called an equivalence class. The term “equivalent” refers to a certain relationship known as an equivalence relation. Any two items are equivalent if they have an equivalence relationship. An equals sign can be used to define a simple equivalence class. We may state that a set of all x such that x = a’ is the equivalence class of a.

Put another way, any equal objects in the set belong to the defined equivalence class.This set appears to be very uninteresting at first glance. However, there are other equivalence relations that make things more intriguing.

Equivalence Class Characteristics

Let’s use the symbol ~ to denote our equivalence connection. If and only if, the relationship is an equivalence relationship:

• It’s self-reflexive: each a in X must be equal to its own; we can write this as and 

a ~ a.

• It’s symmetric: Assume that a and b are in X. If a and b are equivalent, then b is identical to a. This can be written as a ~ b, b ~ a.

• It is transitive: Assume that a, b, and c are X items. If a is equivalent to b and b is equivalent to c, then a must be equivalent to c. This can be written as: for a, b, and c in X; if a ~ b and b ~ c, a ~ c follows.

We may construct the concept of an equivalence class of an element once we’ve double-checked that our connection satisfies the three conditions listed above.

“The equivalence class of a set comprises the set of all x in X such that a and x are related by ~ to each other,” as in “the equivalence classes comprises the set of all x in X such that a and x are correlated by to each other.”

Equivalence Relation Examples

‘Is equal to (=)’ is an equivalence relation on every set of integers A, as we have a = a, 

a = b ⇒b = a, and a = b, b = c⇒ a = c for all elements a, b, c ∈ A. This means that (=) is transitive, symmetric, and reflexive.

On the set of triangles, ‘is comparable to (~)’ is defined: It’s transitive, symmetric, and reflexive.

‘Has same birthday’ is a condition applied to a group of people: It’s transitive, symmetric, and reflexive.

The equivalence relation ‘Is congruent to’ defined on the triangles is reflexive, symmetric, and transitive.

On the set of integers, ‘congruence modulo n (≡ )’ is defined as follows: It’s transitive, symmetric, and reflexive.

On the set of real numbers, the equivalence relation ‘has the same absolute value’ is reflexive, symmetric, and transitive.

Conclusion:

Any two items are equivalent if they have an equivalence relationship. ‘The set of all x such that x =’ a’ is the equivalence class of a. Put another way, any equal objects in the set belong to the defined equivalence class.

An equivalence relation is a reflexive, symmetric, and transitive binary connection. The canonical example of an equivalence relation is the relation is equivalent to. If two items of the given set belong to the same equivalence class, they are comparable.