Equivalence Relation

In mathematics, an equivalence relation is a kind of binary relation that should be reflexive, symmetric and transitive. The well-known example of an equivalence relation is the “equal to (=)” relation. In other words, two elements of the given set are equivalent to each other if they belong to the same equivalence class. In this article, we will discuss the definition of equivalence relation, proof, properties with many solved examples.
Table of Contents:

• Definition
• Proof
  • Reflexive Property
  • Symmetric Property
  • Transitive Property
• Examples
• Practice Problems
• FAQs

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. The equivalence relation is a relationship on the set which is generally represented by the symbol “∼”.

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.

In terms of equivalence relation notation, it is defined as follows:

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

(i.e) For all x, y, z in set A,

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

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 = 3/9.
• For a given set of triangles, the relation of ‘is similar to (~)’ and ‘is congruent to (≅)’ shows equivalence.
• For a given set of integers, the relation of ‘congruence modulo n (≡)’ shows equivalence.
• The image and domain are the same under a function, which 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 us assume that F is 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.

Equivalence Relation Practice Questions

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.

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