A relation R in S = {1, 2, 3} is defined as R = {(1, 1),(2,2),(1,2),(3,3)}. Which of the following elements (s) must be added to make R an equivalence relation ?

Options

A. (2,1),(2,3)
B. (3,2),(2,3)
C. (2,1)
D. (1,3)

Answer

To determine which elements must be added to the relation R = {(1, 1), (2, 2), (1, 2), (3, 3)} in order to make it an equivalence relation, we need to check the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

### Step 1: Check for Reflexivity
A relation is reflexive if every element in the set S is related to itself. For the set S = {1, 2, 3}, we need the pairs (1, 1), (2, 2), and (3, 3) to be in R.

- **Current pairs in R**: (1, 1), (2, 2), (3, 3)
- **Conclusion**: R is reflexive as all required pairs are present.

**Hint**: Verify that each element in the set has a corresponding pair in the relation.

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

- **Current pairs in R**: (1, 2) is present, but (2, 1) is not.
- **Conclusion**: To make R symmetric, we must add (2, 1).

**Hint**: For each pair (a, b) in R, check if the reverse pair (b, a) is also present.

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

- **Current pairs in R**: We have (1, 2) and (2, 2) (which is reflexive) but we need to check if there are any pairs that lead to a new pair.
- Since we are adding (2, 1) to make it symmetric, we also need to check if adding this will affect transitivity.
- With (1, 2) and (2, 1) added, we would also need (1, 1) which is already present.

**Conclusion**: The transitive property is satisfied with the current pairs and the addition of (2, 1).

### Final Conclusion
To make R an equivalence relation, we need to add the pair (2, 1).

### Answer
The element that must be added to make R an equivalence relation is (2, 1).