In a set of ants in a locality, two ants are said to be related iff they walk on a same straight line, then the relation is

Options

A. reflexive and symmetric
B. symmetric and transitive
C. reflexive and transitive
D. equivalence

Answer

To determine the nature of the relation defined for ants walking on the same straight line, we will analyze the properties of reflexivity, symmetry, and transitivity step by step.

### Step 1: Understanding the Relation
We define a relation \( R \) on a set of ants such that two ants \( a \) and \( b \) are related (denoted as \( (a, b) \in R \)) if they walk on the same straight line.

### Step 2: Checking Reflexivity
A relation is reflexive if every element is related to itself. In this case, for any ant \( a \):
- Since ant \( a \) is walking on a straight line, it is indeed related to itself.
- Therefore, \( (a, a) \in R \) for all ants \( a \).

**Conclusion**: The relation is reflexive.

### Step 3: Checking Symmetry
A relation is symmetric if for any two elements \( a \) and \( b \), if \( (a, b) \in R \), then \( (b, a) \in R \) must also hold.
- If ant \( a \) and ant \( b \) are walking on the same straight line, then it follows that ant \( b \) is also walking on the same straight line as ant \( a \).
- Thus, if \( (a, b) \in R \), then \( (b, a) \in R \).

**Conclusion**: The relation is symmetric.

### Step 4: Checking Transitivity
A relation is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \) must also hold.
- If ant \( a \) and ant \( b \) are walking on the same straight line, and ant \( b \) and ant \( c \) are also walking on the same straight line, then it follows that ant \( a \) and ant \( c \) must also be walking on the same straight line.
- Hence, if \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).

**Conclusion**: The relation is transitive.

### Final Conclusion
Since the relation is reflexive, symmetric, and transitive, we conclude that the relation defined for ants walking on the same straight line is an **equivalence relation**.

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