Question

The relation R in the set A = {1, 2, 3, 4} is given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)} is :

• A. Reflexive and symmetric but not transitive
• B. Reflexive and transitive but not symmetric
• C. Symmetric and transitive but not reflexive
• D. an equivalence relation

Answer 1

The Correct Option is B

To determine the properties of the given relation \( R \) on the set \( A = \{1, 2, 3, 4\} \), we need to verify if it is reflexive, symmetric, and transitive.

1. Reflexivity: A relation \( R \) on set \( A \) is reflexive if for every element \( a \in A \), the pair \( (a, a) \in R \).
Checking for each element:

• (1, 1) is in \( R \)
• (2, 2) is in \( R \)
• (3, 3) is in \( R \)
• (4, 4) is in \( R \)

All elements are present, hence \( R \) is reflexive.

2. Symmetry: A relation \( R \) is symmetric if for every pair \((a, b) \in R\), the pair \((b, a) \in R\) must also be in \( R \).
Checking symmetric pairs:

• (1, 2) is in \( R \) but (2, 1) is not.

Therefore, \( R \) is not symmetric.

3. Transitivity: A relation \( R \) is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \) must also be in \( R \).
Checking required transitivity pairs:

• (1, 2) and (2, 2) implies (1, 2), present.
• (1, 2) and (2, 3) does not exist, not applicable.
• (1, 3) and (3, 2) implies (1, 2), present.

All applicable conditions are satisfied, hence \( R \) is transitive.

Given these checks, the relation \( R \) is reflexive and transitive but not symmetric.