Question

Let a set A = A₁ ∪ A₂ ∪ ... ∪ A_k, where A_i ∩ A_j = Φ for i ≠ j, 1 ≤ i, j ≤ k. Define the relation R from A to A by R = {(x, y): y ∈ A_i if and only if x ∈ A_i, 1 ≤ i ≤ k}. Then, R is ______.

विकल्प

• reflexive, symmetric but not transitive

• reflexive, transitive but not symmetric

• reflexive but not symmetric and transitive

• an equivalence relation

Answer 1

Let a set A = A₁ ∪ A₂ ∪ ... ∪ A_k, where A_i ∩ A_j = Φ for i ≠ j, 1 ≤ i, j ≤ k. Define the relation R from A to A by R = {(x, y): y ∈ A_i if and only if x ∈ A_i, 1 ≤ i ≤ k}. Then, R is an equivalence relation.

Explanation:

Set A = A₁ ∪ A₂ ∪ A₃... ∪ A_x, where A_i ∩ A_j = Φ; i ≠ j, 1 ≤ i, j ≤ k

And relation R = {(x, y) : y ∈ A_i : iff x ∈A_i; 1 ≤ i ≤ k}

(1) Symmetric: If (x, y) ∈ R, then (y, x) ∈ R

∴ Given relation is symmetric.

(2) Reflexive: ∵ (a, a) ∈ R for all a ∈ A_i

∴ Given relation is reflexive

(3) Transitive: If (x, y) ∈ R and (y, z) ∈ R

⇒ y ∈ A; iff x ∈ A_i and z ∈ A_i iff y ∈ A_i

⇒ z ∈ A; iff x

⇒ A_i

⇒ (x, z) ∈ R

∴ Given relation is transitive.

Since, given relation is symmetric, reflexive and transitive

∴ It is an equivalence relation.