To determine the properties of the relation \( R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\} \) defined on the set \( A = \{1, 2, 3, 4\} \), we will check if it is a function, reflexive, symmetric, and transitive.
### Step 1: Check if \( R \) is a function
A relation is a function if every element in the domain (set \( A \)) maps to exactly one element in the codomain.
- For \( 1 \): maps to \( 3 \)
- For \( 2 \): maps to \( 4 \) and \( 3 \) (two images)
- For \( 3 \): maps to \( 1 \)
- For \( 4 \): maps to \( 2 \)
Since the element \( 2 \) has two images (4 and 3), \( R \) is **not a function**.
### Step 2: Check if \( R \) is reflexive
A relation is reflexive if every element in the set \( A \) relates to itself, i.e., \( (a, a) \) is in \( R \) for all \( a \in A \).
- Check for \( (1, 1) \): not present
- Check for \( (2, 2) \): not present
- Check for \( (3, 3) \): not present
- Check for \( (4, 4) \): not present
Since none of the pairs \( (1, 1), (2, 2), (3, 3), (4, 4) \) are present in \( R \), it is **not reflexive**.
### Step 3: Check if \( R \) is symmetric
A relation is symmetric if for every \( (a, b) \in R \), the pair \( (b, a) \) is also in \( R \).
- Check \( (1, 3) \): \( (3, 1) \) is present
- Check \( (4, 2) \): \( (2, 4) \) is present
- Check \( (2, 4) \): \( (4, 2) \) is present
- Check \( (2, 3) \): \( (3, 2) \) is not present
- Check \( (3, 1) \): \( (1, 3) \) is present
Since \( (2, 3) \) is present but \( (3, 2) \) is not, \( R \) is **not symmetric**.
### Step 4: Check if \( R \) is transitive
A relation is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \) must also be in \( R \).
- Check pairs:
- From \( (1, 3) \) and \( (3, 1) \): \( (1, 1) \) is not present.
- From \( (2, 3) \) and \( (3, 1) \): \( (2, 1) \) is not present.
- From \( (4, 2) \) and \( (2, 4) \): \( (4, 4) \) is not present.
- From \( (2, 4) \) and \( (4, 2) \): \( (2, 2) \) is not present.
Since there are several instances where the transitive property fails, \( R \) is **not transitive**.
### Conclusion
The relation \( R \) is:
- Not a function
- Not reflexive
- Not symmetric
- Not transitive
