Equivalence Relations

An equivalence Relation is a type of relation that satisfies three fundamental properties: reflexivity, symmetry, and transitivity. These properties ensure that it defines a partition on a set, where elements are grouped into equivalence classes based on their similarity or equality.

• It is essential in various mathematical and theoretical contexts, including algebra, set theory, and graph theory.
• It provides a structured way to compare and classify elements within a set.

The equivalence relation is a relationship on the set which is generally represented by the symbol “∼”. An equivalence relation on a set is a binary relation that satisfies three fundamental properties:

• Reflexivity: ∀ a ∈ S: a ~ a
• Symmetry: ∀ a, b ∈ S: a ~ b ⇒ b ~ a
• Transitivity: ∀ a, b, c ∈ S: (a ~ b) ∧ (b ~ c) ⇒ a ~ c

Reflexive Relation

A relation R on a set A is called a reflexive relation if

(a, a) ∈ R ∀ a ∈ A, i.e. aRa for all a ∈ A, where R is a subset of (A ✕ A), i.e. the cartesian product of set A with itself.

This means if element “a” is present in set A, then a relation “a” to “a” (aRa) should be present in relation R. If any such aRa is not present in R, then R is not a reflexive relation.

Symmetric Relation

A relation R on a set A is called a symmetric relation if and only if

∀ a, b ∈ A, if (a, b) ∈ R then (b, a) ∈ R and vice versa i.e., where R is a subset of (A x A), i.e. the cartesian product of set A with itself.

This means if an ordered pair of elements “a” to “b” (aRb) is present in relation R, then an ordered pair of elements “b” to “a” (bRa) should also be present in relation R. If any such bRa is not present for any aRb in R, then R is not a symmetric relation.

Transitive Relation

A relation R on a set A is called a transitive relation if and only if

∀ a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R, where R is a subset of (A x A), i.e. the cartesian product of set A with itself.

This means if an ordered pair of elements “a” to “b” (aRb) and “b” to “c” (bRc) is present in relation R, then an ordered pair of elements “a” to “c” (aRC) should also be present in the relation R. If any such aRc is not present for any aRb & bRc in R R is not a transitive relation.

Example of Equivalence Relation

A classic example of an equivalence relation is the relation of "equality" on the set of real numbers. Given any two real numbers "a" and "b":

• Reflexivity: "a = a" is always true for any real number "a."
• Symmetry: If "a = b," then "b = a."
• Transitivity: If "a = b" and "b = c," then "a = c."

Some other examples include:

• Congruence (in modular arithmetic)
• Congruence of Geometric Shapes
• Equivalence of Parallel Lines

Properties of Equivalence Relation

• Equivalence relations are often denoted by the symbol "≡" or by writing "∼" between related elements.
• An example of an equivalence relation is the "congruence modulo n" relation in modular arithmetic, where two integers are related if their difference is a multiple of n. This relation is reflexive, symmetric, and transitive.
• Equivalence relations are widely used in mathematics, computer science, and other fields for classifying objects, defining partitions, and simplifying complex problems.

How to Verify an Equivalence Relation?

Let us assume that R is 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?

To prove that R is an equivalence relation, we must show that R is

• Reflexive Relation
• Symmetric Relation
• Transitive Relation

Let's verify all these relations for any given relation R.

Verify Reflexive Relation

The process of identifying if any given relation is reflexive:

• Check for the existence of every aRa tuple in the relation for all a present in the set.
• If every tuple exists, then the relation is reflexive. Otherwise, not reflexive.

Example for Reflexive Relation

Consider set A = {a, b} and a relation R = {{a, a}, {b, b}}.

For the element a in A: 
⇒ The pair {a, a} is present in R.
⇒ Hence aRa is satisfied.

For the element b in A:
⇒ The pair {b, b} is present in R.
⇒ Hence bRb is satisfied.

As the condition for 'a', ‘b’ is satisfied, the relation is reflexive.

Verify Symmetric Relation

To verify a symmetric relation, do the following:

• Manually check for the existence of every bRa tuple for every aRb tuple in the relation.
• If any of the tuples do not exist, then the relation is not symmetric; else, it is symmetric.

Follow the example given below for better understanding.

Example for Symmetric Relation

Consider set A = { 1, 2, 3, 4 } and a relation R = { (1, 2), (1, 3), (2, 1), (3, 4), (3, 1),(4.3) }

For the pair (1, 2) in R:

⇒ The reversed pair (2, 1) is present in the relation.
⇒ This pair satisfies the condition

For the pair (1, 3) in R:

⇒ The reversed pair (3, 1) is present in the relation.
⇒ This pair satisfies the condition

For the pair (2, 1) in R:

⇒ The reversed pair (1, 2) is present in the relation.
⇒ This pair satisfies the condition

For the pair (3, 4) in R:

⇒ The reversed pair (4, 3) is present in the relation.
⇒ This pair satisfy the condition

For the pair (3, 1) in R:

⇒ The reversed pair (1, 3) is present in the relation
⇒ This pair satisfies the condition

As the set satisfies the condition, the relation is symmetric.

Verify Transitive Relation

To verify the transitive relation:

• Firstly, find the tuples of the form aRb & bRc in the relation.
• For every such pair, check if aRc is also present in R.
• If any of the tuples do not exist, then the relation is not transitive; else, it is transitive.

Example for Transitive Relation

Consider set R = {(1, 2), (2, 3), (3, 4), (1, 3), (2, 4), (1, 4)}

For the pairs (1, 2) and (2, 3):
⇒ The relation (1, 3) exists.
⇒ This satisfies the condition.

For the pairs (2, 3) and (3, 4):
⇒ The relation (2, 4) exists.
⇒ This satisfies the condition.

For the pairs (1, 3) and (3, 4):
⇒ The relation (1, 4) exists.
⇒ This satisfies the condition.

All possible pairs satisfy the condition of transitivity.

So, the given relation is transitive.

Hence, the transitive property is proved.

Related Articles:

• Types of Function
• One to One Function
• Injection Functions

Solved Problems on Equivalence Relation

Question 1: Show that the relation R, defined in the set A of all polygons as R = {(P1, P2): P1 and P2 have the same number of sides}, is an equivalence relation on A.

Solution:

Given A = set of all polygons
R = {(P1, P2): P1 and P2 have same number of sides}

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

• Reflexive: Let P A
  Clearly, Number of sides of P = number of sides of P
  (P, P) ⋿ R ∀ P ⋿ A
  Hence, R is reflexive.

• Symmetric: Let P1, P2 ⋿ A
  Let (P1, P2) ⋿ R ⇒ P1 and P2 have same number of sides
  ⇒ P2 and P1 have same number of sides
  ⇒ (P2, P1) ⋿ A
  Hence R is Symmetric.

• Transitive: Let P1, P2 ⋿ A
  Let (P1, P2) ⋿ R and (P2, P3) ⋿ R
  ⇒ Number of sides of P1 = number of sides of P2 and
  ⇒ Number of sides of P2 = number of sides of P3
  ⇒ Number of sides of P1 = number of sides of P3
  ⇒ (P1, P3) ⋿ R
  Hence R is transitive.

Thus, R is reflexive, symmetric and transitive and hence R is an equivalence relation on A.

Question 2: Prove that a relation defines an equivalence relation for triangles in geometry.

Solution:

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

• Reflexive: Every triangle is similar to itself.
  x is similar to x, ∀ x ⋿ R ⇒ xRx, ∀ x ⋿ T
  so, R is reflexive on T.

• Symmetric: x is similar to y
  ⇒ y is similar to x.
  ⇒ yRx
  Hence R is symmetric relation on R.

• Transitive: x is similar to y and y is similar to z
  ⇒ xRy and yRz
  ⇒ x is similar to z
  ⇒xRz.
  Hence R is transitive relation on R.

Hence R is an equivalence relation on T.

Unsolved Questions on Equivalence Relation

Question 3: Show that the relation R in the set A = {x ⋿ Z : 0 ≤ x ≤ 12} given by R = {(a, b): a = b} is an equivalence relation.

Question 4: Let a relation R be defined on the set Z of integers by x R y <⇒ x = y; x, y ⋿ Z. Show that R is an equivalence relation.

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

Question 2: Let f: x→y be a function. Define a relation R in X as R = {(a, b): f(a) = f(b)}. Examine if R is an equivalence relation.