If A = {1,2,3} and R = { (1,1}, ( 2,2), (3,3)} then R is reflexive, symmetric or transitive?

Options

A. reflexive and symmetric
B. symmetric and transitive
C. reflexive and transitive
D. equivalence

Explanation

To determine whether the relation \( R \) is reflexive, symmetric, or transitive, we will analyze each property step by step.

### Given:
- Set \( A = \{1, 2, 3\} \)
- Relation \( R = \{(1, 1), (2, 2), (3, 3)\} \)

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if every element \( a \) in set \( A \) is related to itself, meaning \( (a, a) \in R \) for all \( a \in A \).

- Check for \( 1 \): \( (1, 1) \in R \)
- Check for \( 2 \): \( (2, 2) \in R \)
- Check for \( 3 \): \( (3, 3) \in R \)

Since all elements of \( A \) are related to themselves, we conclude that \( R \) is reflexive.

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

- The pairs in \( R \) are \( (1, 1), (2, 2), (3, 3) \).
- For \( (1, 1) \), \( (1, 1) \) is also in \( R \).
- For \( (2, 2) \), \( (2, 2) \) is also in \( R \).
- For \( (3, 3) \), \( (3, 3) \) is also in \( R \).

Since all pairs are of the form \( (a, a) \), we can say that \( R \) is symmetric.

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

- The pairs in \( R \) are \( (1, 1), (2, 2), (3, 3) \).
- Since all pairs are of the form \( (a, a) \), if we take any \( (a, b) \) and \( (b, c) \), we can see that \( a = b = c \).
- Therefore, \( (a, c) \) will also be of the form \( (a, a) \) which is in \( R \).

Thus, \( R \) is transitive.

### Conclusion
The relation \( R \) is reflexive, symmetric, and transitive.