In the set `A = {1, 2, 3, 4, 5}`, a relation R is defined by `R = {(x, y) \ x, y in A and x < y}`. Then R is

Options

A. Reflexive
B. Symmetric
C. Transitive
D. Equivalence relation.

Explanation

To determine the properties of the relation \( R \) defined on the set \( A = \{1, 2, 3, 4, 5\} \) by \( R = \{(x, y) \mid x, y \in A \text{ and } x < y\} \), we will check if \( R \) is reflexive, symmetric, transitive, or an equivalence relation.

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

- For \( R \) to be reflexive, we need \( (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) \) to be in \( R \).
- However, since \( x < y \), there are no pairs of the form \( (a, a) \) because \( a \) cannot be less than itself.

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

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

- If \( (x, y) \in R \), then \( x < y \).
- For \( R \) to be symmetric, we would need \( (y, x) \) to also be in \( R \), which means \( y < x \).
- This is not possible since if \( x < y \), then \( y < x \) cannot hold true.

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

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

- Let’s take \( (x, y) \in R \) such that \( x < y \) and \( (y, z) \in R \) such that \( y < z \).
- Then, by the transitive property of inequalities, we have \( x < y < z \) which implies \( x < z \).
- Therefore, \( (x, z) \in R \).

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

### Step 4: Determine if \( R \) is an Equivalence Relation
An equivalence relation must be reflexive, symmetric, and transitive.

- Since we have established that \( R \) is not reflexive and not symmetric, it cannot be an equivalence relation.

### Final Conclusion
The relation \( R \) is transitive but not reflexive or symmetric. Therefore, the correct answer is that \( R \) is a transitive relation.

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