Educational

Check Whether the Statements are True or False for the Relation R = {(a, b) : a = b²} Check Whether the Statements are True or False Question Let \(R\) be a relation from \(N\) to \(N\) defined by \[ R=\{(a,b):a,b\in N \text{ and } a=b^2\} \] Are the following statements true? (i) \[ (a,a)\in […]

Find the Total Number of Relations from A into B Find the Total Number of Relations from \(A\) into \(B\) Question Let \[ A=\{x,y,z\} \] and \[ B=\{a,b\}. \] Find the total number of relations from \(A\) into \(B\). Solution \[ n(A)=3,\quad n(B)=2 \] Number of elements in \[ A\times B \] is \[ n(A\times

List All Relations on A = {a, b} and Find Their Number List All Relations on \(A=\{a,b\}\) and Find Their Number Question Let \[ A=\{a,b\}. \] List all relations on \(A\) and find their number. Solution \[ A\times A= \{ (a,a),(a,b),(b,a),(b,b) \} \] A relation on \(A\) is any subset of \[ A\times A. \]

Determine the Domain and Range of the Relation S = {(a, b) : b = |a − 1|} Determine the Domain and Range of the Relation Question Determine the domain and range of the relation \[ S=\{(a,b):b=|a-1|,\ a\in Z,\ |a|\le3\} \] Solution Since \[ |a|\le3, \] \[ a=-3,-2,-1,0,1,2,3 \] Now, \[ b=|a-1| \] \[ |-3-1|=4

Determine the Domain and Range of the Relation R = {(a, b) : a < 5, b = 4} Determine the Domain and Range of the Relation Question Determine the domain and range of the relation \[ R=\{(a,b):a\in N,\ a

Determine the Domain and Range of the Relation R = {(x, x³)} Determine the Domain and Range of the Relation Question Determine the domain and range of the relation \[ R=\{(x,x^3):x \text{ is a prime number less than }10\} \] Solution Prime numbers less than \(10\) are \[ 2,3,5,7 \] Therefore, \[ R= \{ (2,8),(3,27),(5,125),(7,343)

Determine the Domain and Range of the Relation R = {(x, x+5)} Determine the Domain and Range of the Relation Question Determine the domain and range of the relation \[ R=\{(x,x+5):x\in\{0,1,2,3,4,5\}\} \] Solution The relation is \[ R= \{ (0,5),(1,6),(2,7), \] \[ (3,8),(4,9),(5,10) \} \] Domain = set of first elements \[ \boxed{ \{0,1,2,3,4,5\} }

Find the Total Number of Relations from A into B Find the Total Number of Relations from \(A\) into \(B\) Question Let \[ A=\{1,2\} \] and \[ B=\{3,4\}. \] Find the total number of relations from \(A\) into \(B\). Solution \[ n(A)=2,\quad n(B)=2 \] Number of elements in \[ A\times B \] is \[ n(A\times

Show That R is an Empty Relation from A into B Show That \(R\) is an Empty Relation from \(A\) into \(B\) Question Let \[ A=\{3,5\} \] and \[ B=\{7,11\}. \] Let \[ R=\{(a,b):a\in A,\ b\in B,\ a-b \text{ is odd}\}. \] Show that \(R\) is an empty relation from \(A\) into \(B\). Solution Elements

Express R and R⁻¹ as Sets of Ordered Pairs | Relation Defined by x + 2y = 8 Express \(R\) and \(R^{-1}\) as Sets of Ordered Pairs Question Let \(R\) be a relation in \(N\) defined by \[ (x,y)\in R \iff x+2y=8. \] Express \(R\) and \(R^{-1}\) as sets of ordered pairs. Solution Given, \[