1.

Let A = {1, 2, 3}, and let R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R2 = {(2, 2), (3, 1), (1, 3)}, R3 = {(1, 3), (3, 3)}. Find whether or not each of the relations R1, R2, R3 on A is(i) reflexive(ii) symmetric(iii) transitive.

Answer»

Consider as R1

Given that R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}

Reflexivity:

Here, (1, 1), (2, 2), (3, 3) ∈ R

Clearly, R1 is reflexive.

Symmetry:

Here, (2, 1) ∈ R1,

But (1, 2) ∉ R1

So, R1 is not symmetric.

Transitivity:

Here, (2, 1) ∈ R1 and (1, 3) ∈ R1,

But (2, 3) ∉ R1

So, R1 is not transitive.

Now we consider R2

Given that R2 = {(2, 2), (3, 1), (1, 3)}

Reflexivity:

Clearly, (1, 1) and (3, 3) ∉ R2

So, R2 is not reflexive.

Symmetry:

Here, (1, 3) ∈ R2 and (3, 1) ∈ R2

Clearly, R2 is symmetric.

Transitivity:

Here, (1, 3) ∈ R2 and (3, 1) ∈ R2

But (3, 3) ∉ R2

So, R2 is not transitive.

Consider as R3

Given that R3 = {(1, 3), (3, 3)}

Reflexivity:

Clearly, (1, 1) ∉ R3

So, R3 is not reflexive.

Symmetry:

Here, (1, 3) ∈ R3, but (3, 1) ∉ R3

So, R3 is not symmetric.

Transitivity:

Here, (1, 3) ∈ R3 and (3, 3) ∈ R3

Also, (1, 3) ∈ R3

Clearly, R3 is transitive.


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