Let `A={1,2,3,4,5}andR` be a relation defined by `R={(x,y):x,y in A,x+y=5}`. The ,R is

Options

A. reflexive and symmetric but not transitive
B. an equivalence relation
C. symmetric but neither reflexive nor transitive
D. neither reflexive nor symmetric but transitive

Answer

To solve the problem, we need to analyze the relation \( R \) defined on the set \( A = \{1, 2, 3, 4, 5\} \) where \( R = \{(x, y) : x, y \in A, x + y = 5\} \).

### Step 1: Identify the pairs in relation \( R \)

We need to find all pairs \( (x, y) \) such that \( x + y = 5 \) and both \( x \) and \( y \) belong to the set \( A \).

- If \( y = 1 \), then \( x = 5 - 1 = 4 \) → Pair: \( (4, 1) \)
- If \( y = 2 \), then \( x = 5 - 2 = 3 \) → Pair: \( (3, 2) \)
- If \( y = 3 \), then \( x = 5 - 3 = 2 \) → Pair: \( (2, 3) \)
- If \( y = 4 \), then \( x = 5 - 4 = 1 \) → Pair: \( (1, 4) \)
- If \( y = 5 \), then \( x = 5 - 5 = 0 \) (not in \( A \))

Thus, the relation \( R \) consists of the following pairs:
\[ R = \{(1, 4), (2, 3), (3, 2), (4, 1)\} \]

### Step 2: Check if \( R \) is reflexive

A relation is reflexive if every element \( a \in A \) is related to itself, i.e., \( (a, a) \in R \) for all \( a \in A \).

- Check pairs: \( (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) \)
- None of these pairs are in \( R \).

**Conclusion**: \( R \) is **not reflexive**.

### Step 3: Check if \( R \) is symmetric

A relation is symmetric if for every \( (x, y) \in R \), the pair \( (y, x) \) is also in \( R \).

- Check pairs:
- \( (1, 4) \) is in \( R \) and \( (4, 1) \) is also in \( R \).
- \( (2, 3) \) is in \( R \) and \( (3, 2) \) is also in \( R \).

**Conclusion**: \( R \) is **symmetric**.

### Step 4: Check if \( R \) is transitive

A relation is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \) must also be in \( R \).

- Check pairs:
- From \( (1, 4) \) and \( (4, 1) \), we do not have \( (1, 1) \) in \( R \).
- From \( (2, 3) \) and \( (3, 2) \), we do not have \( (2, 2) \) in \( R \).

**Conclusion**: \( R \) is **not transitive**.

### Final Conclusion

The relation \( R \) is:
- Not reflexive
- Symmetric
- Not transitive

Thus, the answer is that \( R \) is symmetric but neither reflexive nor transitive.