The minimum number of elements that must be a | vedclass

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(B) Given $R = \{(a, b), (b, c)\}$ on the set $A = \{a, b, c\}$.For $R$ to be symmetric,if $(x, y) \in R$,then $(y, x) \in R$.Adding elements for symm

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$7$

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(B) Given $R = \{(a, b), (b, c)\}$ on the set $A = \{a, b, c\}$.For $R$ to be symmetric,if $(x, y) \in R$,then $(y, x) \in R$.Adding elements for symmetry:Since $(a, b) \in R$,we must add $(b, a)$.Since $(b, c) \in R$,we must add $(c, b)$.Now $R = \{(a, b), (b, a), (b, c), (c, b)\}$.For $R$ to be transitive,if $(x, y) \in R$ and $(y, z) \in R$,then $(x, z) \in R$.Using $(a, b) \in R$ and $(b, c) \in R$,we must add $(a, c)$.Since $(a, c) \in R$,for symmetry,we must add $(c, a)$.Now $R = \{(a, b), (b, a), (b, c), (c, b), (a, c), (c, a)\}$.Checking transitivity again:$(a, b) \in R$ and $(b, a) \in R \Rightarrow (a, a) \in R$.$(b, c) \in R$ and $(c, b) \in R \Rightarrow (b, b) \in R$.$(a, c) \in R$ and $(c, a) \in R \Rightarrow (c, c) \in R$.Adding these,$R = \{(a, b), (b, a), (b, c), (c, b), (a, c), (c, a), (a, a), (b, b), (c, c)\}$.The elements added are $(b, a), (c, b), (a, c), (c, a), (a, a), (b, b), (c, c)$.Total elements added $= 7$.

The minimum number of elements that must be added to the relation $R = \{(a, b), (b, c)\}$ on the set $\{a, b, c\}$ so that it becomes symmetric and transitive is:

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