Let `R={(1,3),(4,2),(2,4),(2,3),(3,1)}` be a relation on the set `A={1,2,3,4}`. Then relation R is

Options

A. a function
B. transitive
C. not symmetric
D. reflexive

Answer

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

### Step 1: Check if \( R \) is Reflexive
A relation \( R \) is reflexive if every element in the set \( A \) is related to itself. This means we need to check if the pairs \( (1,1) \), \( (2,2) \), \( (3,3) \), and \( (4,4) \) are in \( R \).

- **Check**:
- \( (1,1) \) is not in \( R \)
- \( (2,2) \) is not in \( R \)
- \( (3,3) \) is not in \( R \)
- \( (4,4) \) is not in \( R \)

Since none of the pairs are present, \( R \) is **not reflexive**.

### Step 2: Check if \( R \) is Symmetric
A relation \( R \) is symmetric if for every pair \( (x,y) \) in \( R \), the pair \( (y,x) \) is also in \( R \).

- **Check**:
- For \( (1,3) \), \( (3,1) \) is in \( R \) (symmetric)
- For \( (4,2) \), \( (2,4) \) is in \( R \) (symmetric)
- For \( (2,4) \), \( (4,2) \) is in \( R \) (symmetric)
- For \( (2,3) \), \( (3,2) \) is not in \( R \) (not symmetric)
- For \( (3,1) \), \( (1,3) \) is in \( R \) (symmetric)

Since \( (2,3) \) does not have \( (3,2) \) in \( R \), \( R \) is **not symmetric**.

### Step 3: Check if \( R \) is Transitive
A relation \( R \) is transitive if whenever \( (x,y) \) and \( (y,z) \) are in \( R \), then \( (x,z) \) must also be in \( R \).

- **Check**:
- From \( (1,3) \) and \( (3,1) \), we would need \( (1,1) \) which is not in \( R \).
- From \( (2,3) \) and \( (3,1) \), we would need \( (2,1) \) which is not in \( R \).
- From \( (2,4) \) and \( (4,2) \), we would need \( (2,2) \) which is not in \( R \).
- From \( (4,2) \) and \( (2,3) \), we would need \( (4,3) \) which is not in \( R \).

Since we found pairs that violate the transitive property, \( R \) is **not transitive**.

### Step 4: Check if \( R \) is a Function
A relation \( R \) is a function if every element in the domain (set \( A \)) maps to exactly one element in the codomain.

- **Check**:
- For \( 1 \), it maps to \( 3 \) (unique).
- For \( 2 \), it maps to \( 4 \) and \( 3 \) (not unique).
- For \( 3 \), it maps to \( 1 \) (unique).
- For \( 4 \), it maps to \( 2 \) (unique).

Since \( 2 \) maps to two different elements, \( R \) is **not a function**.

### Conclusion
The relation \( R \) is:
- Not reflexive
- Not symmetric
- Not transitive
- Not a function