An equivalence relation is a sort of binary relation that should be reflexive, symmetric plus transitive. In set theory, a relation is defined as a way of showing a connection between any two sets. A relation in mathematics defines the link between two distinct sets of information. If two sets are considered, the relation between them will be confirmed if there is an association between the elements of the given sets.

There are 8 main types of relations which involve: empty relation, identity relation, universal relation, symmetric relation, transitive relation, equivalence relation, inverse relation and reflexive relation.

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Equivalence Relation Definition

Equivalence Relation is a sort of binary relation that should be reflexive, symmetric plus transitive in nature. The well-known instance of an equivalence relation is the “equal to (=)” relation. In other words, we can consider when two elements of the provided set are equivalent to each other if they relate to the same equivalence class.

Condition for equivalence:

• If the relation R is reflexive, then all the elements of the given set are mapped with itself, such that for all a∈Q, then (a, a)∈R.
• The relation R is said to be symmetric on set Q, if (a,b)∈R, then (b, a)∈R, such that a,b∈Q.
• The relation R on set Q, if (a,b)∈R and (b,c)∈R, then (a,c)∈R for all a,b,c ∈R is said to be a transitive relation.

Equivalence relation example:

If A = {3, 4, 5}, then relation R = {(3, 3), (4, 4), (5, 5), (3, 5), (5, 3), (3, 4), (4, 5)} is an equivalence relation ∵ the relation R is reflexive, symmetric and transitive as shown above respectively.

In terms of mathematical concepts, a binary relation over the sets A and B denotes a subset of the cartesian product A × B consisting of components of the form (a, b) such that a ∈ A and b ∈ B. If any of the three conditions that are reflexive, symmetric or transitive is not supported, the relation cannot be an equivalence one. An equivalence type of relation is commonly expressed by the symbol ‘~’.

The properties of equivalence relations include reflexive, symmetric, and transitive. These ensure that every element relates to itself, mutual relationships hold in both directions, and consistency is maintained across linked elements. Together, they define a balanced and logical way to group related elements.

Reflexive Property

The reflexive property means that every element is related to itself.

So, if we take any number a from a set A, the pair (a, a) must be in the relation R.

Now, for positive integers, we can write:
((a, b), (a, b)) ∈ R

This is true because multiplying the same numbers gives the same result:
ab = ab for all positive integers.

So, the reflexive property is satisfied.

Symmetric Property

The symmetric property says that if one pair is in the relation, then its reverse must also be in the relation.

In other words, if (a, b) ∈ R, then (b, a) must also be in R.

So, for the relation, if ((a, b), (c, d)) ∈ R, then ((c, d), (a, b)) must also belong to R.

Given that ((a, b), (c, d)) ∈ R, it means ad = bc.
And because multiplication is commutative (order doesn’t matter), cb = da.

So, we also have ((c, d), (a, b)) ∈ R.

Hence, the symmetric property is true.

Transitive Property

The transitive property means that if the first pair is related to the second, and the second is related to the third, then the first is related to the third.

So if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

For positive integers, if we have:
((a, b), (c, d)) ∈ R and ((c, d), (e, f)) ∈ R,
we must check if ((a, b), (e, f)) ∈ R also holds.

From the given,
ad = cb and cf = de

This means:
a/b = c/d and c/d = e/f,
So, clearly: a/b = e/f,
which gives af = be

Thus, ((a, b), (e, f)) ∈ R.

Therefore, the transitive property is also true.

Important Notes on Equivalence Relation 

• An equivalence relation is a special rule used to compare elements in a set.
   

• For a relation to be called an equivalence relation, it must follow three rules:
  Reflexive (every element is related to itself),
  Symmetric (if one element is related to another, then the reverse is also true), and
  Transitive (if one is related to a second, and the second to a third, then the first is related to the third).
   

• Equivalence relations group elements into equivalence classes, where all members of the same group are considered equal or related to each other.
   

• These groups don’t overlap, and every element in the set belongs to one and only one equivalence class.

Proving a Relation is Not an Equivalence Relation

We already know how to prove if a relation is an equivalence relation. Now, let’s look at an example of a relation that is not one.

Consider a relation R on the set of all integers. The rule is: (a, b) is in R if and only if a is greater than or equal to b (written as a ≥ b).

Let’s check the three important properties:

• Reflexive:
  Every number is equal to itself, so a ≥ a is true for any integer a.
  This means the relation is reflexive.

• Symmetric:
  If a ≥ b, does that mean b ≥ a? Not always.
  For example, 12 ≥ 9 is true, but 9 ≥ 12 is false.
  So, the relation is not symmetric.

• Since the relation is not symmetric, there’s no need to check the third property (transitive). If any one property fails, the relation is not an equivalence relation.

Proof of Equivalence Relation

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

i.e. for all p, q, r in set X:

p ∼ p (Reflexivity).

p ∼ q if and only if q ∼ p (Symmetry).

If p∼q and q∼r, then p∼r (Transitivity).

Check out this article on Sequences and Series.

To learn how to prove if a relation is an equivalence one and determine how many equivalence relations are on a set; let us consider an example.

• Specify a relation P on the set of natural numbers N as (x, y) ∈ P if and only if x = y. Now, we will confirm that the relation P is reflexive, symmetric and transitive.
• Reflexive Property – Since each natural number is equivalent to itself, that is, x = x for all x ∈ N ⇒ (x, x) ∈ P for all x ∈ N. Therefore, P is reflexive.
• Symmetric Property – For x, y ∈ N, let (x, y) ∈ P ⇒ x = y ⇒ y = x ⇒ (y, x) ∈ P. Since x, y are arbitrary, P is symmetric in nature.
• Transitive Property – For x, y, z ∈ N, let (x, y) ∈ P and (y, z) ∈ P ⇒ x = y and y = z ⇒ x = z (as numbers similar to the same number are equivalent to one another) ⇒ (x, z) ∈ P. Since x, y, z are arbitrary, p is transitive.

As P, explained on the set of natural numbers N, is reflexive, symmetric, plus transitive, and P is an equivalence relation.

Equivalence Relation Examples

The equivalence relationships can be explained in terms of the following examples:

• The symbol of ‘is equal to (=)’ on a set of numbers/ characters/ symbols.
• For example: 1/4 = 2/8
•  For a set A as for all elements p, q, r ∈ A, we have p = p, p = q ⇒ q = p, and p = q, q = r ⇒ p = r. This implies (=) is reflexive, symmetric and transitive.
• For an assigned set of triangles, the relation of ‘is similar to (denoted by “~”)’ and ‘is congruent to (≅)’ confirms equivalence that is they are reflexive, symmetric and transitive.
• ‘Congruence modulo n (≡)’ is defined on the set of integers: It is reflexive, symmetric, and transitive that is it shows equivalence.
• The image and domain are equivalent under a function, which confers the relation of equivalence.
• For a set of all angles, ‘possesses the same cosine’.
• ‘Has the same absolute value’ specified on the set of real numbers shows equivalence as it is reflexive, symmetric, and transitive.

Example 1: Prove that the relation R is an equivalence type in the set P= { 3, 4, 5,6 } given by the relation R = { (p, q):|p-q| is even }.

Solution:

R = { (p, q):|p-q| is even }. Where p, q belongs to P.

Reflexive Property :

From the provided relation |p – p| = | 0 |=0.

• And 0 is always even.
• Therefore, |p – p| is even.
• Hence, (p, p) relates to R

So R is Reflexive.

Symmetric Property :

From the given relation |p – q| = |q – p|.

• We know that |p – q| = |-(q – p)|= |q – p|
• Hence |p – q| is even.
• Next |q – p| is also even.
• Accordingly, if (p, q) ∈ R, then (q, p) also belongs to R.

Therefore R is symmetric.

Transitive Property :

• If |p – q| is even, then (p-q) is even.
• Similarly, if |q-r| is even, then (q-r) is also even.
• The summation of even numbers is too even.
• So, we can address it as p – q+ q-r is even.
• Next, p – r is further even.

Accordingly,

• |p – q| and |q-r| is even, then |p – r| is even.
• Consequently, if (p, q) ∈ R and (q, r) ∈ R, then (p, r) also refers to R.
• Therefore R is transitive.

Learn more about Probability with this article.

Example 2: Consider A = {2, 3, 4, 5} and R = {(5, 5), (5, 3), (2, 2), (2, 4), (3, 5), (3, 3), (4, 2), (4, 4)}.

Confirm that R is an equivalence type of relation.

Reflexive: Relation R is reflexive because (5, 5), (2, 2), (3, 3) and (4, 4) ∈ R.

Symmetric: Relation R is symmetric as whenever (a, b) ∈ R, (b, a) also relates to R.

Example: (3, 5) ∈ R ⟹ (5, 3) ∈ R.

Transitive: Relation R is transitive as whenever (a, b) and (b, c) relate to R, (a, c) also relates to R.

Example: (3, 5) ∈ R and (5, 3) ∈ R ⟹ (3, 3) ∈ R.

Accordingly, R is reflexive, symmetric and transitive. So, R is an Equivalence Relation.

Check more topics of Mathematics here.

Here is a brief summary of the various types of relations in discrete mathematics along with the representation:

Important Points on Equivalence Relation

An equivalence type of relation signifies a binary relation established on a set X such that the relation is reflexive, symmetric and transitive.

• The equivalence type of relation distributes the set into disjoint equivalence classes.
• All elements relating to the same equivalence class are equal to each other.

Some of the terms associated with an equivalence relation are: Take an equivalence relation R defined on a set A including a, b ∈ A.

• Equivalence Class – An equivalence class is a subset B of A in such a way that (a, b) ∈ R for all a, b ∈ B and a, b cannot exist outside of B.
• Mathematically, an equivalence class of a is expressed as [a] = {x ∈ A: (a, x) ∈ R} which includes all elements of A which are linked to ‘a’. All elements of A equivalent to each other refer to the same equivalence class.
• Partition – A partition of set A implies a non-empty set of disjoint subsets of A such that none of the elements of A is in two subsets of A and elements relating to the identified subset are related to each other. The union of subsets in the partition is equivalent to set A.
• Quotient Set – A quotient set denotes a set of all equivalence classes of an equivalence type of relation expressed by A/R = {[a]: a ∈ A}.

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