Let A = {1, 2, 3, 4} and R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2),

1.	Let A = {1, 2, 3, 4} and R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (1, 3), (3, 2)}.Show R is reflexive and transitive but not symmetric.
Answer» We have relation R on set A = {1, 2, 3, 4} which is defined as R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (1, 3), (3, 2)}. Since, (1, 1), (2, 2), (3, 3), (4, 4) ∈ R. Therefore, (a, a) R, ∀a ∈ A = {1, 2, 3, 4}. Hence, R is a reflexive relation on set A. Now, (1, 2) ∈ R but (2, 1) ∉ R. Therefore, relation R is not symmetric. Now, (1, 3) ∈ R, (3, 2) ∈ R and (1, 2) ∈ R. Therefore, if there is any (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R for any a, b and c ∈ A. Therefore, relation R is transitive relation. Hence, relation R is reflexive and transitive but not symmetric.

Let A = {1, 2, 3, 4} and R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (1, 3), (3, 2)}.Show R is reflexive and transitive but not symmetric.

Answer»

We have relation R on set A = {1, 2, 3, 4} which is defined as R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (1, 3), (3, 2)}.

Since, (1, 1), (2, 2), (3, 3), (4, 4) ∈ R.

Therefore, (a, a) R, ∀a ∈ A = {1, 2, 3, 4}.

Hence, R is a reflexive relation on set A.

Now, (1, 2) ∈ R but (2, 1) ∉ R.

Therefore, relation R is not symmetric.

Now, (1, 3) ∈ R, (3, 2) ∈ R and (1, 2) ∈ R.

Therefore, if there is any (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R for any a, b and c ∈ A.

Therefore, relation R is transitive relation.

Hence, relation R is reflexive and transitive but not symmetric.

Discussion

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