LetR be the relation on the seM = {1, 2, 3, 4} given by R = {(1,2), (2,2), (1,1), (4,4), (1,3), (3, 3), (3,2)}. Then

Options

A. R is reflexive and symmetric but not transitive
B. R is reflexive and transitive but not symmetric
C. R is symmetric and transitive but not reflexive
D. R is an equivalence relation

Answer

To determine the properties of the relation \( R \) on the set \( M = \{1, 2, 3, 4\} \) given by \( R = \{(1,2), (2,2), (1,1), (4,4), (1,3), (3,3), (3,2)\} \), we will check if the relation is reflexive, symmetric, and transitive.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if for every element \( a \) in the set \( M \), the pair \( (a, a) \) is in \( R \).

- The elements of \( M \) are \( 1, 2, 3, 4 \).
- We check if \( (1,1), (2,2), (3,3), (4,4) \) are in \( R \):
- \( (1,1) \) is in \( R \)
- \( (2,2) \) is in \( R \)
- \( (3,3) \) is in \( R \)
- \( (4,4) \) is in \( R \)

Since all pairs are present, the relation \( R \) is reflexive.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if whenever \( (a, b) \) is in \( R \), then \( (b, a) \) is also in \( R \).

- We check each pair in \( R \):
- For \( (1,2) \), \( (2,1) \) is not in \( R \) (not symmetric).
- For \( (2,2) \), \( (2,2) \) is in \( R \) (symmetric).
- For \( (1,1) \), \( (1,1) \) is in \( R \) (symmetric).
- For \( (4,4) \), \( (4,4) \) is in \( R \) (symmetric).
- For \( (1,3) \), \( (3,1) \) is not in \( R \) (not symmetric).
- For \( (3,3) \), \( (3,3) \) is in \( R \) (symmetric).
- For \( (3,2) \), \( (2,3) \) is not in \( R \) (not symmetric).

Since there are pairs for which the symmetric condition fails, the relation \( R \) is not symmetric.

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( (a, b) \) and \( (b, c) \) are in \( R \), then \( (a, c) \) must also be in \( R \).

- We check the pairs:
- From \( (1,2) \) and \( (2,2) \), we have \( (1,2) \) in \( R \) (transitive).
- From \( (1,3) \) and \( (3,2) \), we have \( (1,2) \) in \( R \) (transitive).
- From \( (3,2) \) and \( (2,2) \), we have \( (3,2) \) in \( R \) (transitive).
- From \( (1,2) \) and \( (2,2) \), we have \( (1,2) \) in \( R \) (transitive).
- From \( (1,3) \) and \( (3,3) \), we have \( (1,3) \) in \( R \) (transitive).
- From \( (1,3) \) and \( (3,2) \), we have \( (1,2) \) in \( R \) (transitive).

Since all necessary conditions for transitivity are satisfied, the relation \( R \) is transitive.

### Conclusion
- \( R \) is reflexive.
- \( R \) is not symmetric.
- \( R \) is transitive.

Thus, the correct statement about the relation \( R \) is that it is reflexive and transitive but not symmetric.