Question

Let A = {1, 2, 3}. Give an example of a relation on A which is

1. Symmetric but neither reflexive nor transitive.

2. Transitive but neither reflexive nor symmetric.

Answer 1

1. R = {(1, 2),(2, 1)}

(1, 1) ∉ R ⇒ R is not reflexive

(1, 2) ∈ R ⇒ (2, 1) ∈ R, R is symmetric

(1, 2) ∈ R, (2, 1) ∈ R but (1,1) ∉ R,

R is not transitive

2. R = {(1, 2),(1, 3),(2, 3)}

(1, 1) ∉ R ⇒ R is not reflexive

(1, 2) ∈ R ⇒ but (2,1) ∉ R, R is not symmetric

(1, 2) ∈ R, (2, 3) ∈R ⇒ (1, 3) ∈ R,

R is transitive.