Let $R = \{(1,2),(2,3),(3,3)\}$ be a relation defined on the set $\{1,2,3,4\}$. Then the minimum number of elements, needed to be added in $R$ so that $R$ becomes an equivalence relation, is:
Step-by-Step Solution
Key Concept: For R to be an equivalence relation it must be reflexive, symmetric, and transitive. Add all missing reflexive pairs first, then symmetric pairs, then close under transitivity.
Add reflexive pairs: $(1,1),(2,2),(4,4)$ (already has $(3,3)$). Add symmetric: $(2,1),(3,2)$. By transitivity, $(1,2)+(2,3)\Rightarrow(1,3)$; add $(1,3)$ and $(3,1)$. Total added: $(1,1),(2,2),(4,4),(2,1),(3,2),(1,3),(3,1)$ = 7 elements.
<div class="key-concept"><strong>Key Concept:</strong> For R to be an equivalence relation it must be reflexive, symmetric, and transitive. Add all missing reflexive pairs first, then symmetric pairs, then close under transitivity.</div>
<div class="trap-box"><strong>Trap:</strong> After adding reflexive pairs $(1,1),(2,2),(4,4)$ and symmetric pairs $(2,1),(3,2)$, check transitivity: $(1,2)$ and $(2,3)$ are in R so $(1,3)$ must be added, and then its symmetric pair $(3,1)$ too.</div>
Correct Answer: 7