If A = {1, 2, 3, 4 }, define relations on
Question:

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.

Solution:

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

(i) Let R1 = {(1, 1), (1, 2), (1, 3), (2, 3), (2, 2), (1, 3), (3, 3)}

R1 is reflexive as (1, 1), (2, 2) and (3, 3) lie is R1.

R1 is transitive as (1, 2) ∈ R1, (2, 3) ∈ R1 ⇒ (1, 3) ∈ R1

Now, (1, 2) ∈ R1 ⇒ (2, 1) ∉ R1.

(ii) Let R2 = {(1, 2), (2, 1)}

Now, (1, 2) ∈ R2, (2, 1) ∈ R2

So, it is symmetric,

And, clearly R2 is not reflexive as (1, 1) ∉ R2

Also, R2 is not transitive as (1, 2) ∈ R2, (2, 1) ∈ R2 but (1, 1) ∉ R2

(iii) Let R3 = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

R3 is reflexive as (1, 1) (2, 2) and (3, 3) ∈ R1

R3 is symmetric as (1, 2), (1, 3), (2, 3) ∈ R1 ⇒ (2, 1), (3, 1), (3, 2) ∈ R1

Therefore, Ris reflexive, symmetric and transitive.

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