Equivalence Relation

In mathematics, relations and functions are the most important concepts. In class 11 and class 12, we have studied the important ideas which are covered in the relations and function. The concepts are used to solve the problems in different chapters like probability, differentiation, integration, and so on. In this article, let us discuss one of the concepts called “Equivalence Relation” with its definition, proofs, different properties along with the solved examples.
Table of Contents:

Equivalence Relation Definition

A relation R on a set A is said to be an equivalence relation if and only if the relation R is reflexive, symmetric and transitive.

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

Symmetric: A relation is said to be symmetric, if (a, b) ∈ R, then (b, a) ∈ R.

Transitive: A relation is said to be transitive if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

Equivalence relations can be explained in terms of the following examples:

  • The sign of ‘is equal to’ on a set of numbers, for example, 1/3 is equal to 3/9.
  • For a given set of triangles, the relation of ‘is similar to’ and ‘is congruent to’.
  • For a given set of integers, the relation of ‘is congruent to, modulo n’ shows equivalence.
  • The image and domain are the same under a function, shows the relation of equivalence.
  • For a set of all angles, ‘has the same cosine’.
  • For a set of all real numbers, ‘ has the same absolute value’.

Equivalence Relation Proof

Here is an equivalence relation example to prove the properties.

Let us assume that R be a relation on the set of ordered pairs of positive integers such that ((a,b), (c,d))∈ R if and only if ad=bc. Is R an equivalence relation?

In order to prove that R is an equivalence relation, we must show that R is reflexive, symmetric and transitive.

The Proof for the given condition is given below :

Reflexive Property

According to the reflexive property, if (a, a) ∈ R, for every a∈A

For all pairs of positive integers,

((a,b),(a,b))∈ R.

Clearly, we can say

ab = ab for all positive integers.

Hence, the reflexive property is proved.

Symmetric Property

From the symmetric property,

if (a, b) ∈ R, then we can say (b, a) ∈ R

For the given condition,

if ((a,b),(c,d)) ∈ R, then ((c,d),(a,b)) ∈ R.

If ((a,b),(c,d))∈ R, then ad = bc and cb = da

since multiplication is commutative.

Therefore ((c,d),(a,b)) ∈ R

Hence symmetric property is proved.

Transitive Property

From the transitive property,

if (a, b) ∈ R and (b, c) ∈ R, then (a, c) also belongs to R

For the given set of ordered pairs of positive integers,

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

then ((a,b),(e,f) ∈ R.

Now, assume that ((a,b), (c,d))∈ R and ((c,d), (e,f)) ∈ R.

Then we get, ad = cb and cf = de.

The above relation implies that a/b = c/d and that c/d = e/f,

so a/b = e/f we get af = be.

Therefore ((a,b),(e,f))∈ R.

Hence transitive property is proved.

Equivalence Relation Examples

Go through the equivalence relation examples and solutions provided here

Question 1 :

Let assume that F be a relation on the set R real numbers defined by xFy if and only if x-y is an integer. Prove that F is an equivalence relation on R.

Solution :

Reflexive: Consider x belongs to R,then x – x = 0 which is an integer. Therefore xFx.

Symmetric: Consider x and y belongs to R and xFy. Then x – y is an integer. Thus, y – x = – ( x – y), y – x is also an integer. Therefore yFx.

Transitive: Consider x and y belongs to R, xFy and yFz. Therefore x-y and y-z are integers. According to the transitive property, ( x – y ) + ( y – z ) = x – z is also an integer. So that xFz.

Thus, R is an equivalence relation on R.

Question 2:

Show that the relation R is an equivalence relation in the set A = { 1, 2, 3, 4, 5 } given by the relation R = { (a, b):|a-b| is even }.

Solution :

R = { (a, b):|a-b| is even }. Where a, b belongs to A

Reflexive Property :

From the given relation,

|a – a| = | 0 |=0

And 0 is always even.

Thus, |a-a| is even

Therefore, (a, a) belongs to R

Hence R is Reflexive

Symmetric Property :

From the given relation,

|a – b| = |b – a|

We know that |a – b| = |-(b – a)|= |b – a|

Hence |a – b| is even,

Then |b – a| is also even.

Therefore, if (a, b) ∈ R, then (b, a) belongs to R

Hence R is symmetric

Transitive Property :

If |a-b| is even, then (a-b) is even.

Similarly, if |b-c| is even, then (b-c) is also even.

Sum of even number is also even

So, we can write it as a-b+ b-c is even

Then, a – c is also even

So,

|a – b| and |b – c| is even , then |a-c| is even.

Therefore, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) also belongs to R

Hence R is transitive.

Practice problems on Equivalence Relation

Solve the practise problems on the equivalence relation given below:

  1. Prove that the relation R is an equivalence relation, given that the set of complex numbers is defined by z1 R z2 ⇔[(z1-z2)/(z1+z2)] is real.
  2. Show that the given relation R is an equivalence relation, which is defined by (p, q) R (r, s) ⇒ (p+s)=(q+r)
  3. Check the reflexive, symmetric and transitive property of the relation x R y, if and only if y is divisible by x, where x, y ∈ N.

Frequently Asked Questions on Equivalence Relation

What is meant by equivalence relation?

In mathematics, the relation R on the set A is said to be an equivalence relation, if the relation satisfies the properties, such as reflexive property, transitive property, and symmetric property.

What are the three different properties of the equivalence relation?

The three different properties of equivalence relation are:
Reflexive Property
Symmetric Property
Transitive Property

Explain reflexive, transitive and symmetric property.

A relation R is said to be reflective, if (x,x) ∈ R, for every x ∈ set A
A relation R is said to be symmetric, if (x,y) ∈ R, then (y, x) ∈ R
A relation R is said to be transitive, if (x, y) ∈ R and (y,z)∈ R, then (x, z) ∈ R

Can we say the empty relation is an equivalence relation?

We can say that the empty relation on the empty set is considered as an equivalence relation. But, the empty relation on the non-empty set is not considered as an equivalence relation.

Can we say every relation is a function?

No, every relation is not considered as a function, but every function is considered as a relation.

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