Let R be a relation on R , given by R = {(a,b):3a - 3b + is an irrational number }. Then R is: Relation (Mathematics)

(c) : $R\{(a,b):3a - 3b +$ is an irrational number $\}\\$ Reflexive : For $(a,a)$ , we have $3a - 3a + \\=$ which is an irrational . number. $\\ \therefore R$ is reflexive Symmetric : $\\$ Let $(a,b) \in R$ i.e., $3a - 3b +$ is an irrational number $\\$ Now, we need to check $(b,a) \in R$ or not. $\\$ Let $3a =$ and $3b = 8$ then $3a - 3b+ \\= - 8 + = 2 - 8,\\$ which is an irrational . . number but $3b - 3a + \\= 8 - + = 8,\\$ which is not an irrational $\therefore$ For $(a,b) \in R,(b,a) \notin R$ $\\ \therefore R$ is not symmetric $\\$ Transitive : Let $(a,b)$ and $(b,c) \in R\\$ Let $3a = 8,3b = 2,3c =$ then, $3a - 3b + \\= 8 - 2 + = 8 -$ , which is an irrational also, $3b - 3c + \\= 2 - + = 2,\\$ which is an irrational but $3a - 3c + = 8 - + = 8,\\$ which is not an irrational $\Rightarrow (a,c) \notin R\\$ $\therefore R$ is not transitive.

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

Let R be a relation on Z X Z defined by (a,b)R(c,d) if and only if ad - bc is divisible by 5 . Then R is

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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