Equivalence Class and Relation

A relation R on a (non-empty) set A is called an equivalence relation iff it is :

(i) reflexive, (ii) symmetric and (iii) transitive, i.e. iff

1. a R a for all a ∈ A
2. a R b implies b R a for all a, b ∈ A and
3. a R b, b R c implies a R c for all a, b, c ∈ A

If R is an equivalence relation on a set A, then the equivalence class of an element \[a \in A\] is the set of all elements of A related to a.

This means an equivalence class collects all elements that are considered “equivalent” under the given relation.

Let T be the set of all triangles in a plane, with R a relation in T given by

R = {(T₁, T₂) : T₁ is congruent to T₂}. Show that R is an equivalence relation.

Solution:
R is reflexive, since every triangle is congruent to itself.

Further, (T₁, T₂) ∈ R ⇒ T₁ is congruent to T₂ ⇒ T₂ is congruent to T₁ ⇒ (T₂, T₁) ∈ R. Hence, R is symmetric.

Moreover, (T₁, T₂), (T₂, T₃) ∈ R ⇒ T₁ is congruent to T₂ and T₂ is congruent to T₃ ⇒ T₁ is congruent to T₃ ⇒ (T₁, T₃) ∈ R.

Therefore, R is an equivalence relation.

Let A = {real numbers}

Let R = {(a, b) : a, b ∈ A and a − b < 5}

Is R an equivalence relation? Justify your answer.

Solution:

(i) Reflexive: For all a ∈ A, a − a = 0 < 5 ⇒ aRa.

∴ R is reflexive.

(ii) Symmetric: For example, let a = 2, b = 8, then

a − b = 2 − 8 = −6 < 5 ⇒ aRb

But, b − a = 8 − 2 = 6 ≮ 5 ⇒ bRa.

Hence, R is not symmetric.

(iii) Transitive: If a = 4, b = 0, c = −4, then

a − b = 4 − 0 = 4 < 5 ⇒ aRb

and b − c = 0 − (−4) = 4 < 5 ⇒ bRc

but a − c = 4 − (−4) = 8 ≮ 5 ⇒ aRc

∴ R is not transitive.

⇒ R is not an equivalence relation.

Let A = {2, 4, 6, 8} and R be the relation ‘is greater than’ on the set A. Write R as a set of ordered pairs.

1. reflexive?
2. symmetric?
3. equivalence relation?

Justify your answer.

Solution:

Given A = {2, 4, 6, 8} and R is the relation ‘is greater than’ on A, therefore,

R = {(8, 6), (8, 4), (8, 2), (6, 4), (6, 2), (4, 2)}

1. R is not reflexive, because (2, 2) ∉ R
2. R is not symmetric, because (8, 6) ∈ R but (6, 8) ∉ R
3. R is not an equivalence relation, because R is an equivalence relation only when it is reflexive, symmetric and transitive.

However, the above relation is transitive, because if x > y and y > z, then x > z.

• An equivalence relation must be reflexive, symmetric, and transitive.

• Equality is a standard example of an equivalence relation.

• Relations like “greater than” or “is mother of” are not equivalence relations because they do not satisfy all three properties.

• Equivalence classes are sets of mutually related elements.

• Equivalence relations and partitions of sets are closely connected.