Suppose X = {1,2,3,4} and R is a relation on X. If R = {(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)}, then which one of the following is correct?

Options

A. A) R is reflexive and symmetric, but not transitive
B. B) R is symmetric and transitive, but not reflexive
C. C) R is reflexive and transitive, but not symmetric
D. D) R is neither reflexive nor transitive, but symmetric

Answer

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

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

- The elements of \( X \) are \( 1, 2, 3, 4 \).
- We need to check if \( (1, 1), (2, 2), (3, 3), (4, 4) \) are in \( R \).
- From the given relation \( R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)\} \), we see:
- \( (1, 1) \) is in \( R \)
- \( (2, 2) \) is in \( R \)
- \( (3, 3) \) is in \( R \)
- \( (4, 4) \) is **not** in \( R \)

Since \( (4, 4) \) is not in \( R \), the relation is **not reflexive**.

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

- We will check each pair in \( R \):
- For \( (1, 1) \), \( (1, 1) \) is in \( R \) (symmetric).
- For \( (2, 2) \), \( (2, 2) \) is in \( R \) (symmetric).
- For \( (1, 2) \), \( (2, 1) \) is also in \( R \) (symmetric).
- For \( (2, 1) \), \( (1, 2) \) is also in \( R \) (symmetric).
- For \( (2, 3) \), \( (3, 2) \) is also in \( R \) (symmetric).
- For \( (3, 2) \), \( (2, 3) \) is also in \( R \) (symmetric).
- For \( (3, 3) \), \( (3, 3) \) is in \( R \) (symmetric).

Since all pairs satisfy the symmetry condition, the relation is **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 will check the pairs:
- From \( (1, 2) \) and \( (2, 3) \), we should have \( (1, 3) \) in \( R \), but \( (1, 3) \) is **not** in \( R \).

Since we found a counterexample, the relation is **not transitive**.

### Conclusion
Based on the checks:
- The relation \( R \) is **not reflexive**.
- The relation \( R \) is **symmetric**.
- The relation \( R \) is **not transitive**.

Thus, the correct option is that \( R \) is symmetric.

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