Question

A class-room teacher is keen to assess the learning of her students the concept of “relations” taught to them. She writes the following five relations each defined on the set A = {1, 2, 3}: R1 = {(2, 3), (3, 2)} R2 = {(1, 2), (1, 3), (3, 2)} R3 = {(1, 2), (2, 1), (1, 1)} R4 = {(1, 1), (1, 2), (3, 3), (2, 2)} R5 = {(1, 1), (1, 2), (3, 3), (2, 2), (2, 1), (2, 3), (3, 2)}

The students are asked to answer the following questions about the above relations:

1. Identify the relation which is reflexive, transitive but not symmetric.
2. Identify the relation which is reflexive and symmetric but not transitive.
   1. Identify the relations which are symmetric but neither reflexive nor transitive.
                                                     OR
   2. What pairs should be added to the relation R₂ to make it an equivalence relation

Answer 1

(i) R₄ = {(1, 1), (1, 2), (3, 3), (2, 2)}

(1, 1), (2, 2), (3, 3) are reflexive, symmetric, and transitive.

(1, 2), (2, 2) are transitive.

But (1, 2) ∈ R

(2, 1) ∉ R

∵ R₄ is not symmetric.

(ii) R₁ = {(2, 3) (3, 2)} Symmetric

R₂ = {(1, 2) (1, 3) (3, 2)} Transitive

R₃ = {(1, 2) (2, 1) (1, 1)} Symmetric and transitive

R₄ = {(1, 1) (1, 2) (3, 3) (2, 2)} Reflexive but not symmetric

R₅ = {(1, 1) (1, 2) (3, 3) (2, 2) (2, 1) (2, 3) (3, 2)}. This is reflexive, symmetric, and transitive

Hence, no, reflection is “Reflexive and symmetric but not transitive”.

(iii) (a) R₁ is symmetric but neither reflexive nor transitive

R₁ = {(2, 3), (3, 2)}

(2, 2), (3, 3) ∉ R₁  ...(It is not reflexive)

and (2, 3), (3, 2) ∈ R but (2, 2) ∉ R₁    ...(So, it is not transitive)

                                               OR

(iii) (b) R₂ = {(1, 2), (1, 3), (3, 2)}

To make R₂ equivalence, we required (1, 1), (2, 2), (3, 3) to make it reflexive and (2, 1), (3, 1), (2, 3) to make it symmetric.

Hence, it is reflexive, symmetric, and transitive 

∴ R₂ = {(1, 1) (2, 2) (3, 3) (1, 2) (2, 1) (1, 3) (3, 1) (3, 2) (2, 3)}