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: 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.

Let A = { 1,3,4,6,9} and B = { 2,4,5,8,10}. Let R be a relation defined on A X B such that R = {( ( a_{1},b_{1} ), ( a_{2},b_{2} )):a_{1} ? b_{2} and b_{1} ? a_{2}}. Then the number of elements in the set R is

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Let A = { 1,2,3,4,...,10} and B = { 0,1,2,3,4}. The number of elements in the relation = {(a,b) in A X A : 2(a - b)^{2} + 3(a - b) in B} is .

The number of relations, on the set { 1,2,3} containing (1,2) and (2,3), which are reflexive and transitive but not symmetric, is .

Let A = { 0,3,4,6,7,8,9,10} and R be the relation defined on A such that R = {(x,y) in A X A:x - y is odd positive integer or x - y = 2}. The minimum number of elements that must be added to the relation R, so that it is a symmetric relation, is equal to .

If R is the smallest equivalence relation on the set { 1,2,3,4} such that {(1,2),(1,3)} subset R, then the number of elements in R is .

Let X = { 1,2,3,4,5}. The number of different ordered pairs ( Y,Z ) that can be formed such that Y subseteq X,Z subseteq X and Y cap Z. pty is

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