Let `A={1,2,3,4,5}` and R be a relation defined by `R={(x,y),x,yinA,x+y=5}`. Then 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 \( A = \{1, 2, 3, 4, 5\} \) such that \( R = \{(x, y) \mid x, y \in A, x + y = 5\} \).

### Step 1: Identify pairs \((x, y)\) such that \(x + y = 5\)

We will find all pairs \((x, y)\) where both \(x\) and \(y\) belong to the set \(A\) and their sum equals 5.

- 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 valid since \(0 \notin A\)

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

### Step 2: Check if the relation is symmetric

A relation is symmetric if for every \((x, y) \in R\), \((y, x) \in R\) as well.

- Check \((4, 1)\): \((1, 4) \in R\)
- Check \((3, 2)\): \((2, 3) \in R\)
- Check \((2, 3)\): \((3, 2) \in R\)
- Check \((1, 4)\): \((4, 1) \in R\)

Since all pairs satisfy the symmetric condition, the relation \( R \) is symmetric.

### Step 3: Check if the relation is reflexive

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

- Check for \(1\): \((1, 1) \notin R\)
- Check for \(2\): \((2, 2) \notin R\)
- Check for \(3\): \((3, 3) \notin R\)
- Check for \(4\): \((4, 4) \notin R\)
- Check for \(5\): \((5, 5) \notin R\)

Since none of the pairs \((x, x)\) are in \(R\), the relation \( R \) is not reflexive.

### Step 4: Check if the relation is transitive

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

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

Since there are instances where the transitive condition does not hold, the relation \( R \) is not transitive.

### Conclusion

The relation \( R \) is symmetric but not reflexive and not transitive. Therefore, the answer is:

**R is symmetric but not reflexive or transitive.**

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