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

The minimum number of elements that must be added to th... | Relation (Mathematics)

(b) : Given relation; $R = \{(a,b),(b,c)\}$ and set $= \{ a,b,c\}\\$ For symmetric, Since $(a,b),(b,c) \in R\\$ So, $(b,a),(c,b)$ be in $R.\\$ For transitive, Since $(a,b),(b,c) in R\\$ So, ( $a,c$ ) should be in $R.\\$ then ( $c,a$ ) should also be in $R\\$ Since $(a,c)$ lies in $R.\\$ Since, $(a,b),(b,a) \in R$ . So, $(a,a) \in R$ should also lie in $R$ . Since $(c,b),(b,c) in R.\\$ So, $(c,c)$ should also lie in $R.\\$ Since $(b,c),(c,b) \in R$ . So, $(b,b)$ should also lie in $R\\$ $\therefore$ Elements to be added are: $\\(b,a),(c,b),(a,c),(c,a),(a,a),(c,c),(b,b).\\$ Total no. of elements to be added $= 7$ .

Explanation

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

Let R be the set of real numbers. end{enumerate} Statement-1: A = {(x,y) in R X R:y - x is an integer } is an equivalence . relation on R. Statement-2: B = {(x,y) in R X R:x = alpha y for some rational number } is . an equivalence relation on R.

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