The relation `R={(1,3),(3,5)}` is defined on the set with minimum number of elements of natural numbers. The minimum number of elements to be included in R so that R is an equivalence relation, is

Options

A. 5
B. 6
C. 7
D. 8

Answer

To determine the minimum number of elements to be included in the relation \( R = \{(1,3), (3,5)\} \) so that it becomes an equivalence relation, we need to ensure that \( R \) satisfies the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

### Step 1: Check for Reflexivity
For a relation to be reflexive, every element in the set must relate to itself. This means we need to include pairs of the form \( (a, a) \) for each element \( a \) in the set.

- The elements present in the relation \( R \) are \( 1, 3, \) and \( 5 \).
- Therefore, we need to add the pairs \( (1, 1), (3, 3), \) and \( (5, 5) \) to ensure reflexivity.

**Pairs to add for reflexivity:** \( (1, 1), (3, 3), (5, 5) \)

### Step 2: Check for Symmetry
For a relation to be symmetric, if \( (a, b) \) is in \( R \), then \( (b, a) \) must also be in \( R \).

- From the existing pairs in \( R \):
- \( (1, 3) \) implies we need to add \( (3, 1) \).
- \( (3, 5) \) implies we need to add \( (5, 3) \).

**Pairs to add for symmetry:** \( (3, 1), (5, 3) \)

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

- We have:
- \( (1, 3) \) and \( (3, 5) \) implies we need to add \( (1, 5) \).
- \( (3, 5) \) and \( (5, 3) \) implies we need to add \( (3, 1) \) (which we already added).
- \( (5, 3) \) and \( (3, 1) \) implies we need to add \( (5, 1) \).

**Pairs to add for transitivity:** \( (1, 5), (5, 1) \)

### Summary of Pairs to Add
Now, let's summarize all the pairs we need to add to make \( R \) an equivalence relation:
1. Reflexive pairs: \( (1, 1), (3, 3), (5, 5) \)
2. Symmetric pairs: \( (3, 1), (5, 3) \)
3. Transitive pairs: \( (1, 5), (5, 1) \)

### Total Unique Pairs
Combining all unique pairs, we have:
- \( (1, 1) \)
- \( (3, 3) \)
- \( (5, 5) \)
- \( (3, 1) \)
- \( (5, 3) \)
- \( (1, 5) \)
- \( (5, 1) \)

### Conclusion
The minimum number of elements to be added to \( R \) is 7.