Question

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

• A. only reflexive and symmetric
• B. reflexive only
• C. only reflexive and transitive
• D. reflexive, symmetric and transitive

Hint:

An equivalence relation must satisfy all three properties: reflexive, symmetric, and transitive.
If a relation is reflexive and symmetric, and has pairs like \( (x, y) \) and \( (y, x) \), it must have \( (x, x) \) and \( (y, y) \) to be transitive.

Answer 1

The Correct Option is D

Concept:

• Reflexive: \( (a, a) \in R \) for every \( a \in A \).
• Symmetric: If \( (a, b) \in R \), then \( (b, a) \in R \).
• Transitive: If \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).

Step 1: Check for Reflexivity
The set is \( A = \{1, 2, 3\} \). For \( R \) to be reflexive, it must contain \( (1,1), (2,2), (3,3) \).
Checking the given set \( R \):
\( (1, 1) \in R \)
\( (2, 2) \in R \)
\( (3, 3) \in R \)
Since all are present, \( R \) is reflexive.

Step 2: Check for Symmetry
Checking pairs where \( a \neq b \):
For \( (1, 3) \in R \), we need \( (3, 1) \in R \). Checking \( R \): Yes, \( (3, 1) \) is present.
Since there are no other such pairs, \( R \) is symmetric.

Step 3: Check for Transitivity
We need to check all combinations \( (a, b) \) and \( (b, c) \):
1. Consider \( (1, 3) \) and \( (3, 3) \): \( (1, 3) \in R \). (Satisfied)
2. Consider \( (1, 3) \) and \( (3, 1) \): \( (1, 1) \in R \). (Satisfied)
3. Consider \( (3, 1) \) and \( (1, 3) \): \( (3, 3) \in R \). (Satisfied)
4. Consider \( (3, 1) \) and \( (1, 1) \): \( (3, 1) \in R \). (Satisfied)
Every possible chain leads to an element already in the set. Thus, \( R \) is transitive.

Step 4: Final Conclusion
Since the relation is reflexive, symmetric, and transitive, it is an equivalence relation.
This corresponds to option (D).