Let R = {(1,2), (2,3), (3,3)} be a relation d | vedclass

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(D) For a relation $R$ on a set $A$ to be an equivalence relation,it must be reflexive,symmetric,and transitive. Given $A = \{1, 2, 3, 4\}$ and $R = \

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$7$

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(D) For a relation $R$ on a set $A$ to be an equivalence relation,it must be reflexive,symmetric,and transitive. Given $A = \{1, 2, 3, 4\}$ and $R = \{(1,2), (2,3), (3,3)\}$. $1$. Reflexivity: For $R$ to be reflexive,$(1,1), (2,2), (3,3), (4,4)$ must be in $R$. Since $(3,3)$ is already present,we must add $(1,1), (2,2), (4,4)$. $2$. Symmetry: Since $(1,2) \in R$,we must add $(2,1)$. Since $(2,3) \in R$,we must add $(3,2)$. $3$. Transitivity: Since $(1,2) \in R$ and $(2,3) \in R$,we must have $(1,3) \in R$. Since $(1,3) \in R$,for symmetry we must add $(3,1)$. Now,check for transitivity with the added elements: $(2,1) \in R$ and $(1,3) \in R \implies (2,3) \in R$ (already present). $(3,2) \in R$ and $(2,1) \in R \implies (3,1) \in R$ (already added). $(1,2) \in R$ and $(2,1) \in R \implies (1,1) \in R$ (already added). The set $R$ now contains: $\{(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (2,3), (3,2), (1,3), (3,1)\}$. The elements added are $(1,1), (2,2), (4,4), (2,1), (3,2), (1,3), (3,1)$. Total elements added = $7$.

Let $R = \{(1,2), (2,3), (3,3)\}$ be a relation defined on the set $A = \{1, 2, 3, 4\}$. Then the minimum number of elements needed to be added to $R$ so that $R$ becomes an equivalence relation is:

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