If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
(a) reflexive, transitive but not symmetric

(b) symmetric but neither reflexive nor transitive

(c) reflexive, symmetric and transitive.

Question

Philoid · Accepted Answer

Given that, A = {1, 2, 3, 4 }

(a) Reflexive, transitive but not symmetric

Let R be a relation defined by

R = {(1,1),(1,2),(1,4),(2,2),(2,3),(3,2),(3,3),(4,2),(4,4)} on set A.

R is reflexive ∵ (1,1),(2,2),(3,3),(4,4) ∈ R

R is transitive ∵ (1,4) ∈ R and (4,2) ∈ R ⇒ (1,2) ∈ R

R is not symmetric ∵ (1,4) ∈ R but (4,1) ∉ R

Hence, R is reflexive, transitive but not symmetric.

(b) Symmetric but neither reflexive nor transitive

Let R be a relation defined by

R = {(1,2),(2,1),(2,3),(3,2)} on set A.

R is not reflexive ∵ (1,1),(2,2),(3,3),(4,4) ∉ R

R is symmetric ∵ (1,2) ∈ R ⇒ (2,1) ∈ R and (2,3) ∈ R ⇒ (3,2) ∈ R

R is not transitive ∵ (1,2) ∈ R and (2,1) ∈ R ⇒ (1,1) ∉ R

Hence, R is symmetric but neither reflexive nor transitive.

(c) Reflexive, symmetric and transitive.

Let R be a relation defined by

R = {(1,1),(1,2),(1,4),(2,1),(2,2),(2,3),(3,2),(3,3),(4,1),(4,4)} on set A.

R is reflexive ∵ (1,1),(2,2),(3,3),(4,4) ∈ R

R is symmetric ∵ (1,2),(1,4),(2,3) ∈ R ⇒ (2,1),(4,1),(3,2) ∈ R

R is transitive ∵ (1,2) ∈ R and (2,1) ∈ R ⇒ (1,1) ∈ R

Hence, R is reflexive, symmetric and transitive.

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:(a) reflexive, transitive but not symmetric(b) symmetric but neither reflexive nor transitive(c) reflexive, symmetric and transitive.