If $R$ is the smallest equivalence relation on $\{1, 2, 3, 4\}$ such that $\{(1, 2), (1, 3)\} \subset R$, then the number of elements in $R$ is _____.
Step-by-Step Solution
Key Concept: Start with $\{(1, 2), (1, 3)\}$. Add diagonal $\{(1, 1), (2, 2), (3, 3), (4, 4)\}$ and symmetric pairs $\{(2, 1), (3, 1)\}$. Closure under transitivity: $(2, 1), (1, 3) \to (2, 3)$ and $(3, 1), (1, 2) \to (3, 2)$. Final: diagonal (4) + $\{(1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)\}$ (6) = 10 elements.
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Correct Answer: 10