If R is the smallest equivalence relation on the set `{1, 2, 3, 4}` such that `{(1,2), (1,3)}sub R`, then the number of elements in R is ______

Options

A. 10
B. 12
C. 8
D. 15

Answer

To find the smallest equivalence relation \( R \) on the set \( \{1, 2, 3, 4\} \) such that \( \{(1,2), (1,3)\} \subset R \), we need to ensure that \( R \) satisfies the properties of an equivalence relation: reflexivity, symmetry, and transitivity.

### Step-by-step solution:

1. **Start with the given pairs**: We know that \( (1,2) \) and \( (1,3) \) are in \( R \).

2. **Add reflexive pairs**: For \( R \) to be reflexive, every element in the set must relate to itself. Therefore, we must include:
\[ (1,1), (2,2), (3,3), (4,4) \]

3. **Add symmetric pairs**: Since \( R \) must be symmetric, we need to add the pairs that correspond to the given pairs:
- From \( (1,2) \), we add \( (2,1) \).
- From \( (1,3) \), we add \( (3,1) \).

Now we have:
\[ (1,2), (2,1), (1,3), (3,1) \]

4. **Add transitive pairs**: We need to check for transitivity:
- Since we have \( (1,2) \) and \( (1,3) \), we can derive \( (2,3) \) and \( (3,2) \) because:
- From \( (1,2) \) and \( (1,3) \), we can conclude that if \( 1 \) relates to both \( 2 \) and \( 3 \), then \( 2 \) must relate to \( 3 \) and vice versa.

Thus, we add:
\[ (2,3), (3,2) \]

5. **Final pairs**: Now we compile all the pairs we have:
- Reflexive pairs: \( (1,1), (2,2), (3,3), (4,4) \)
- Symmetric pairs: \( (1,2), (2,1), (1,3), (3,1), (2,3), (3,2) \)

So, the complete set of pairs in \( R \) is:
\[ R = \{(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (1,3), (3,1), (2,3), (3,2)\} \]

6. **Count the elements**: Now, we count the total number of pairs in \( R \):
- There are 10 pairs in total.

### Conclusion:
The number of elements in \( R \) is \( \boxed{10} \).