Let R be a relation on A = {a, b, c} such that R = {(a, a), (b, b), (c, c)}, then R is

Options

A. Reflexive only
B. Symmetric only
C. Non - transitive
D. Equivalence

Answer

To determine the properties of the relation \( R \) defined on the set \( A = \{a, b, c\} \) where \( R = \{(a, a), (b, b), (c, c)\} \), we will check if \( R \) is reflexive, symmetric, and transitive.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if every element in the set \( A \) is related to itself. This means that for every \( x \in A \), the pair \( (x, x) \) must be in \( R \).

- For \( a \): \( (a, a) \in R \)
- For \( b \): \( (b, b) \in R \)
- For \( c \): \( (c, c) \in R \)

Since all elements \( a, b, c \) are related to themselves, \( R \) is reflexive.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if for every pair \( (x, y) \in R \), the pair \( (y, x) \) must also be in \( R \).

- The pairs in \( R \) are \( (a, a), (b, b), (c, c) \).
- For each of these pairs, the reverse pairs are \( (a, a), (b, b), (c, c) \), which are also in \( R \).

Since the reverse of every pair in \( R \) is also in \( R \), \( R \) is symmetric.

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \) must also be in \( R \).

- The pairs in \( R \) are \( (a, a), (b, b), (c, c) \).
- We can check the combinations:
- For \( (a, a) \) and \( (a, a) \): \( (a, a) \) is in \( R \).
- For \( (b, b) \) and \( (b, b) \): \( (b, b) \) is in \( R \).
- For \( (c, c) \) and \( (c, c) \): \( (c, c) \) is in \( R \).

Since all combinations satisfy the transitive property, \( R \) is transitive.

### Conclusion
Since the relation \( R \) is reflexive, symmetric, and transitive, it is classified as an equivalence relation.

### Final Answer
The relation \( R \) is an equivalence relation.

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