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(D) For a relation $R$ on a set $A = \{a, b, c, d\}$ to be an equivalence relation,it must be reflexive,symmetric,and transitive.$1$. Reflexivity: For

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$13$

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(D) For a relation $R$ on a set $A = \{a, b, c, d\}$ to be an equivalence relation,it must be reflexive,symmetric,and transitive.$1$. Reflexivity: For all $x \in A$,$(x, x) \in R$. Thus,we must add $(a, a), (b, b), (c, c), (d, d)$. ($4$ elements)$2$. Symmetry: If $(x, y) \in R$,then $(y, x) \in R$. Given $(a, b), (b, c), (b, d) \in R$,we must add $(b, a), (c, b), (d, b)$. ($3$ elements)$3$. Transitivity: If $(x, y) \in R$ and $(y, z) \in R$,then $(x, z) \in R$.From $(a, b)$ and $(b, c)$,add $(a, c)$.From $(a, b)$ and $(b, d)$,add $(a, d)$.From $(c, b)$ and $(b, d)$,add $(c, d)$.From $(d, b)$ and $(b, c)$,add $(d, c)$.From $(c, a)$ and $(a, b)$,add $(c, b)$ (already added).From $(d, a)$ and $(a, b)$,add $(d, b)$ (already added).From $(a, c)$ and $(c, b)$,add $(a, b)$ (already present).From $(a, d)$ and $(d, b)$,add $(a, b)$ (already present).From $(c, d)$ and $(d, b)$,add $(c, b)$ (already added).From $(d, c)$ and $(c, b)$,add $(d, b)$ (already added).Also need $(c, a)$ and $(d, a)$ due to symmetry of $(a, c)$ and $(a, d)$. ($2$ elements)The full relation $R$ is $\{(a, a), (b, b), (c, c), (d, d), (a, b), (b, a), (b, c), (c, b), (b, d), (d, b), (a, c), (c, a), (a, d), (d, a), (c, d), (d, c)\}$.Total elements = $16$. Given elements = $3$. Elements to add = $16 - 3 = 13$.

The minimum number of elements that must be added to the relation $R = \{(a, b), (b, c), (b, d)\}$ on the set $\{a, b, c, d\}$ so that it is an equivalence relation,is $.........$

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