Prove That (a,b)∈R₁ and (b,c)∈R₁ ⇒ (a,c)∈R₁ is Not Always True Prove That the Given Statement is Not Always True Question For the relation \(R_1\) defined on \(R\) by \[ (a,b)\in R_1 \iff 1+ab>0 \] Prove that \[ (a,b)\in R_1 \text{ and } (b,c)\in R_1 \Rightarrow (a,c)\in R_1 \] is not true for all \(a,b,c\in […]

Find the Domain and Range of the Relation on Z Find the Domain and Range of the Relation on \(Z\) Question Let \(R\) be the relation on \(Z\) defined by \[ R=\{(a,b):a,b\in Z,\ a-b \text{ is an integer}\} \] Find the domain and range of \(R\). Solution Since \[ a,b\in Z, \] the difference \[

Write the Relation in Set Builder Form and Roster Form Write the Relation in Set Builder Form and Roster Form Question Figure 2.15 shows a relationship between the sets \(P\) and \(Q\). Write the relation in: (i) set builder form (ii) roster form Also find its domain and range. Solution From the diagram, \[ 5\to3,\quad

Relation on A Defined by “b is Exactly Divisible by a” Relation on \(A\) Defined by “\(b\) is Exactly Divisible by \(a\)” Question Let \[ A=\{1,2,3,4,5,6\} \] Let \(R\) be a relation on \(A\) defined by \[ R=\{(a,b):a,b\in A,\ b \text{ is exactly divisible by } a\} \] (i) Write \(R\) in roster form (ii)

Write the Relation R = {(x, x³)} in Roster Form Write the Relation \(R\) in Roster Form Question Write the relation \[ R=\{(x,x^3):x \text{ is a prime number less than }10\} \] in roster form. Solution Prime numbers less than \(10\) are \[ 2,3,5,7 \] Now, \[ 2^3=8 \] \[ 3^3=27 \] \[ 5^3=125 \]

Write the Relation R in Roster Form | Difference Between x and y is Odd Write the Relation \(R\) in Roster Form Question \[ A=\{1,2,3,5\} \] and \[ B=\{4,6,9\} \] Define a relation \(R\) from \(A\) to \(B\) by \[ R=\{(x,y): \text{the difference between } x \text{ and } y \text{ is odd},\ x\in A,\

Relation R Defined by y = x + 5 | Roster Form, Arrow Diagram, Domain and Range Relation \(R\) Defined by \(y=x+5\) Question Define a relation \(R\) on the set \(N\) of natural numbers by \[ R=\{(x,y):y=x+5,\ x \text{ is a natural number less than }4,\ x,y\in N\} \] Depict this relation using: (i) roster

Relation on A Defined by 3x − y = 0 | Domain, Co-domain and Range Relation on \(A\) Defined by \(3x-y=0\) Question Let \[ A=\{1,2,3,\ldots,14\} \] Define a relation on \(A\) by \[ R=\{(x,y):3x-y=0,\ x,y\in A\} \] Depict this relation using an arrow diagram. Write its domain, co-domain and range. Solution Given, \[ 3x-y=0 \]

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