The minimum number of elements that must be added to $R = \{(a, b), (b, c)\}$ on $\{a, b, c\}$ so that $R$ becomes symmetric and transitive is _____.
Step-by-Step Solution
Key Concept: For symmetry: add $(b, a), (c, b)$. For transitivity: $(a, b), (b, c) \to (a, c)$ add; $(c, b), (b, a) \to (c, a)$ add. Now $(a, c), (c, a) \to (a, a)$; $(c, a), (a, b) \to (c, b)\checkmark$; $(b, a), (a, b) \to (b, b)$ add; $(b, a), (a, c) \to (b, c)\checkmark$; $(c, a), (a, c) \to (c, c)$ add. Added: $(b, a), (c, b), (a, c), (c, a), (a, a), (b, b), (c, c) = 7$ elements.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: 7