Equivalence Relations Practice Questions

An equivalence relation is a relation on a set that groups elements based on a common property.

• A relation is called an equivalence relation if it satisfies reflexivity, symmetry, and transitivity.
• Elements that are related to each other belong to the same equivalence class.

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.

Practice Questions

Question 1: 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 2: 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 3: 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 4: 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.