Question

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

• A. 10
• B. 12
• C. 8
• D. 15

Answer 1

Answer: (A)

10

Answer 2

Given set - $\\{1, 2, 3, 4\\}$.

To form the smallest equivalence relation on this set that includes $(1, 2)$ and $(1, 3)$, we need to ensure that $R$ is reflexive, symmetric, and transitive.

Step 1. Reflexive pairs: $(1, 1), (2, 2), (3, 3), (4, 4)$
Step 2. Pairs to satisfy given conditions and transitivity:
- Since $(1, 2) \in R$ and $(1, 3) \in R$, we need $(2, 3) \in R$ for transitivity.
- For symmetry, include $(2, 1), (3, 1), (3, 2)$.

Step 3. Final set of pairs: $R = \\{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)\\}$.

Thus, the number of elements in $R$ is 10.

The Correct Answer is: 10