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 :
- (1) 10
- (2) 8
- (3) 9
- (4) 7
Solution:
Set $A = \{1, 2, 3, 4\}$. Current $R = \{(1, 2), (2, 3), (3, 3)\}$.
For Equivalence, $R$ must be Reflexive, Symmetric, and Transitive.
1. Reflexive: Needs $(a, a) \forall a \in A$.
Missing: $(1, 1), (2, 2), (4, 4)$. (3 elements added)
2. Symmetric: Needs $(b, a)$ if $(a, b) \in R$.
Current pairs to check: $(1, 2), (2, 3)$.
Missing: $(2, 1), (3, 2)$. (2 elements added)
3. Transitive: Needs $(a, c)$ if $(a, b)$ and $(b, c) \in R$.
We have $(1, 2)$ and $(2, 3) \implies$ Need $(1, 3)$. (1 element added)
Now we have added $(1, 3)$, so for symmetry, we must also add $(3, 1)$. (1 element added)
Total elements added:
$\{(1, 1), (2, 2), (4, 4)\} \cup \{(2, 1), (3, 2)\} \cup \{(1, 3)\} \cup \{(3, 1)\}$
Count $= 3 + 2 + 1 + 1 = 7$.
Ans. (4)