In order that a relation R defined on a non-empty set A is an equivalence relation, it is sufficient, if R

Options

A. is reflexive
B. is symmetric
C. is transitive
D. possesses all the above three properties

Explanation

To determine the conditions under which a relation \( R \) defined on a non-empty set \( A \) is an equivalence relation, we need to understand the properties that define equivalence relations. An equivalence relation must satisfy three specific properties:

1. **Reflexivity**: For every element \( a \) in set \( A \), the relation \( R \) must include the pair \( (a, a) \). This means that every element is related to itself.

2. **Symmetry**: For any two elements \( a \) and \( b \) in set \( A \), if \( (a, b) \) is in \( R \), then \( (b, a) \) must also be in \( R \). This means that if one element is related to another, then the second element must be related back to the first.

3. **Transitivity**: For any three elements \( a, b, c \) in set \( A \), if \( (a, b) \) is in \( R \) and \( (b, c) \) is in \( R \), then \( (a, c) \) must also be in \( R \). This means that if one element is related to a second, which is in turn related to a third, then the first element must also be related to the third.

In summary, for a relation \( R \) to be an equivalence relation on a non-empty set \( A \), it must be:

- Reflexive
- Symmetric
- Transitive

Thus, it is sufficient for \( R \) to satisfy all three properties to be considered an equivalence relation.

### Step-by-Step Solution:

1. **Identify the properties of equivalence relations**:
- Reflexivity: \( \forall a \in A, (a, a) \in R \)
- Symmetry: \( \forall a, b \in A, (a, b) \in R \Rightarrow (b, a) \in R \)
- Transitivity: \( \forall a, b, c \in A, (a, b) \in R \land (b, c) \in R \Rightarrow (a, c) \in R \)

2. **Conclude that all three properties must hold**:
- A relation \( R \) is an equivalence relation if it is reflexive, symmetric, and transitive.

3. **Select the correct option**:
- The correct option from the given choices is the one that states \( R \) must possess all three properties.