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.
Equivalence Relations
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.
Table of Content
- What is an Equivalence Relation?
- Equivalence Relation Definition
- Example of Equivalence Relation
- Properties of Equivalence Relation
- How to Verify an Equivalence Relation?