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. Equivalence relations are essential in various mathematical and theoretical contexts, including algebra, set theory, and graph theory, as they provide a structured way to compare and classify elements within a set.
In this article, we will learn about the key properties of equivalence relations, how to identify any relation to be an equivalence relation, and their practical applications in fields such as abstract algebra, discrete mathematics, and data analysis. We'll explore examples and exercises to deepen our understanding of Equivalence Relation.
What is an Equivalence Relation?
A relation R on a set is called an equivalence relation if and only if the relation is reflexive, symmetric, and transitive. The equivalence relation is a relationship on the set which is generally represented by the symbol “∼”.
Equivalence Relation Definition
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
What is Reflexive Relation?
A relation R on a set A is called 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.
What is Symmetric Relation?
A relation R on a set A is called 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.
What is Transitive Relation?
A relation R on a set A is called 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 then R is not a transitive relation.
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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 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
- Refelxive Relation
- Symmetric Relation
- Transitive Relation
Let's verify all these relation for any given relation R.
Verify Reflexive Relation
The process of identifying/verifying 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, only then the relation is reflexive. Otherwise, not reflexive.
Follow the example given below for better understanding.
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 does 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 satisfy the condition, the relation is symmetric.
Verify Transitive Relation
To verify transitive relation:
- Firstly, find the tuples of form aRb & bRc in the relation.
- For every such pair check if aRc is also present in R.
- If any of the tuples does not exist then the relation is not transitive else it is transitive.
Follow the below illustration for a better understanding
Example for Transitive Relation
Consider set R = {(1, 2), (1, 3), (2, 3), (3, 4), (1,4)}
For the pairs (1, 2) and (2, 3):
⇒ The relation (1, 3) exists
⇒ This satisfies the condition.
For the pairs (1, 3) and (3, 4):
⇒ The relation (1, 4) exists.
⇒ This satisfies the condition.
So the relation is transitive.
Hence the transitive property is proved.
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Solved Problems on Equivalence Relation
Problem 1: Show that the relation R, defined in the set A of all polygons as R = {(P1, P2): P1 and P2 have 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.
Problem 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.
Practice Questions for Equivalence Relation
Q1: 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.
Q2: 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.
Q3: 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.
Q4: Let a relation R be defined on set Z of integers by x R y <⇒ x=y; x, y ⋿ Z. Show that R is an equivalence relation.
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