Question

A relation R on set A = {1, 2, 3} defined as R = {(1, 1), (2, 2), (1, 2)} is

• A. Reflexive only
• B. Reflexive and Transitive
• C. Symmetric and Transitive
• D. Transitive only

Hint:

Check the pair (3, 3) for reflexivity, the pair (2, 1) for symmetry, and every chain of pairs for transitivity. Only transitivity holds.

Answer 1

The Correct Option is D

Step 1: Recall the three properties:
A relation R on A is reflexive if \( (a, a) \in R \) for every \( a \in A \).
It is symmetric if \( (a, b) \in R \) always gives \( (b, a) \in R \).
It is transitive if \( (a, b) \in R \) and \( (b, c) \in R \) always give \( (a, c) \in R \).

Step 2: Test reflexivity:
For R to be reflexive it must contain \( (1,1), (2,2), (3,3) \).
The pair \( (3, 3) \) is not in R. So R is NOT reflexive.

Step 3: Test symmetry:
The pair \( (1, 2) \) is in R, but \( (2, 1) \) is not in R. So R is NOT symmetric.

Step 4: Test transitivity:
We check every pair of the form \( (a, b), (b, c) \).
\( (1,1),(1,1) \) gives \( (1,1) \in R \).
\( (1,1),(1,2) \) gives \( (1,2) \in R \).
\( (1,2),(2,2) \) gives \( (1,2) \in R \).
\( (2,2),(2,2) \) gives \( (2,2) \in R \).
Every required pair is present, so R is transitive.

Step 5: Match with the options:
(A) says reflexive only. R is not reflexive, so (A) is wrong.
(B) says reflexive and transitive. R is not reflexive, so (B) is wrong.
(C) says symmetric and transitive. R is not symmetric, so (C) is wrong.
(D) says transitive only. R is transitive and it is neither reflexive nor symmetric, so (D) is right.

Final Answer:
R is transitive only, which is option (D). \[ \boxed{\text{(D) Transitive only}} \]