Exercise 2.2
Page no - 2.25
Type 1
Question 1
(i) Let $A=\{2,4,8\}$. Define $R=\{(2,4),(4,8),(8,16)\}$. Is $R$ a relation on $A$ ?
(ii) Let $A=\{1,2,4\}, B=\{2,4,6\}$. Define $R=\{(2,2),(4,4),(6,6)\}$. Is $R$ a relation from $A$ to $B$ ?
(iii) Let $A=\{1,2,3\}, B=\{2,4\}$. Define $R=\{(1,2),(2,2),(2,4),(3,4)\}$. Is $R$ a relation from $A$ to $B$ ?
(iv) Find a relation $R$ from $A=\{1,2,3,4,5\}$ to $B=\mid 1,2,4\}$ defined by $x R y \Leftrightarrow x<y$
Page no - 2.26
Question 2
Write $R$ as a set of ordered pairs
(i) Let $R=\{(x, y): 2 x+3 y<10, x, y \in N\}$.
(ii) Let $R=\{(x, y):(x, y) \in Z \times Z,(x+y)(y+2004)+1=0\}$.
(iii) Let $R=\left\{(x, y):(x, y) \in Z \times Z .4 x^{2}+9 y^{2}=36\right\}$.
Question 3
Let $R=\{(x, y): x, y \in W, y=2 x-4\}$. If $(a,-2) \in R$ and $\left(4, b^{2}\right) \in R$. find the relation $R_{1}=\{(a, b)\}$.
Question 4
Find a linear relation between the components of the ordered pairs of the relation $R$ given by
Question 5
$R=\{(0,2),(-1,5),(2,-4), \ldots\}$
$A=\{a: a \in N, 2 \leq a<6\}$
$B=\{b: b \in N, 3<b<7\}$
$R=\{(a, b): a \in A, b \in B, a$ and $b$ are coprime?
[Hint : $a$ and $b$ are coprime iff HCF of $a$ and $b=1]$
Type 2
Question 6
Find the domain and range of the following relations :
(i) $\{(1,2),(1,4),(1,6),(1,8)\}$.
(ii) $\left\{\left(x, x^{3}\right): x\right.$ is a prime number less than 10$\}$.
(iii) $\{(x, y): x \in N, x<5, y=3\}$.
(iv) $\{(x, y): x \in N, y \in N$ and $x+y=10\}$.
(v) $|(x, x+5): x \in\{0,1,2,3,4,5\}|$
Question 7
Find the domain and range of the relation :
(i) $\left\{\left(x, \frac{1}{x}\right): 0<x<4\right.$ and $x$ is a natural number $\}$.
(ii) $(x, y): x, y \in N$ and $2 x+y=10\}$.
Question 8
Let $R$ be the relation on the set $N$ of all natural numbers defined by $a+3 b=12$. Find (i) $R$ (ii) Domain of $R$ (iii) Range of $R$.
Type 3
Question 9
Let $A=\{1,2\}$ and $B=(3,4\}$. Find the number of relations from $A$ to $B$.
Question 10
Let A= $\{x, y\}$ List all relations on A. Also find the number of relations on A.
Question 11
(i) Let $A=\{a, b, c\}, B=\{x, y\}$. Find the total number of relations from $A$ to $B$.
(ii) Let $A=\{x, y, z\}$ and $B=(1,2\}$, find the number of relations from $A$ to $B$.
Type 4
Question 12
Let $R=\{(1,-1),(2,0),(3,1),(4,2),(5,3)\}$. Then
(i) Write $R$ in builder form
(ii) Represent $R$ by arrow diagram.
Question 13
Let A = $\{1,2,3,4\}$ B = $\{1,2,3,4,5\}$ Let R be a relation from A to B defined by
$a R b \Leftrightarrow a$ divides $b$.
(i) Represent $R$ by lattice.
(ii) Represent $R$ in tabular form.
Page no -2.27
Question 14
Let $A=\{1,2,3\}, B=\{4,5\}$ and $R$ be a relation from $A$ into $B$ given by $R=\{(2,4),(2,5),(3,5)\}$.
Represent $R$ :
(i) in tabular form.
(ii) by arrow diagram.
Question 15
Let $A=\{1,2,3,4,5,6\}$. Define a relation $R$ from $A$ to $A$ by
$R=\{(x, y): y=x+1\}$
Depict this relation using an arrow diagram. Write down the domain, codomain and range of $R$.
Question 16
Write the relation $R=\left\{\left(x, x^{3}\right): x\right.$ is a prime number less than 10 in roster form.
Question 17
Define a relation $R$ on the sct $N$ of all natural numbers by $R=\{(x, y): y=x+5, x$ is a natural number less than 4$\}$.
Depict this relation using
(i) roster form.
(ii) an arrow diagram
Write down its domain and range.
Question 18
Given figure shows a relation $R$ from set $A$ to set $B$. Write this relation in
(i) set builder form (ii) roster form.
(Diagram to be added)
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