Chapter 2 Relations And Functions EXERCISE 2.2
EXERCISE 2.2
1. Let $A=\{1,2,3, \ldots, 14\}$. Define a relation $R$ from $A$ to $A$ by $R=\{(x, y): 3 x-y=0$, where $x, y \in A\}$. Write down its domain, codomain and range.
Show Answer
Answer :
The relation $R$ from $A$ to $A$ is given as
$ R=\{(x, y): 3 x-y=0$, where $x, y \in A \}$
i.e., $R=\{(x, y): 3 x=y$, where $ x, y \in A \}$
$\therefore R=\{(1,3),(2,6),(3,9),(4,12)\}$
The domain of $R$ is the set of all first elements of the ordered pairs in the relation.
$\therefore$ Domain of $R=\{1,2,3,4\}$
The whole set $A$ is the codomainof the relation $R$.
$\therefore$ Codomain of $R=A=\{1,2,3, \ldots, 14\}$
The range of $R$ is the set of all second elements of the ordered pairs in the relation.
$\therefore$ Range of $R=\{3,6,9,12\}$
2. Define a relation $R$ on the set $\mathbf{N}$ of natural numbers by $R=\{(x, y): y=x+5$, $x$ is a natural number less than $4 ; x, y \in \mathbf{N}\}$. Depict this relationship using roster form. Write down the domain and the range.
Show Answer
Answer :
$ R= \{(x, y): y=x+5, x$ is a natural number less than $ 4, x, y \in \mathbf{N} \}$
The natural numbers less than 4 are 1,2 , and 3 .
$\therefore R=\{(1,6),(2,7),(3,8)\}$
The domain of $R$ is the set of all first elements of the ordered pairs in the relation.
$\therefore$ Domain of $R=\{1,2,3\}$
The range of $R$ is the set of all second elements of the ordered pairs in the relation.
$\therefore$ Range of $R=\{6,7,8\}$
3. $A=\{1,2,3,5\}$ and $B=\{4,6,9\}$. Define a relation $R$ from $A$ to $B$ by $R=\{(x, y)$ : the difference between $x$ and $y$ is odd; $x \in A, y \in B\}$. Write $R$ in roster form.
Show Answer
Answer :
$A=\{1,2,3,5\}$ and $B=\{4,6,9\}$
$R=\{(x, y)$ : the difference between $x$ and yis odd; $x \in A, y \in B\}$
$\therefore R=\{(1,4),(1,6),(2,9),(3,4),(3,6),(5,4),(5,6)\}$
4. The Fig2.7 shows a relationship between the sets $P$ and $Q$. Write this relation
(i) in set-builder form (ii) roster form. What is its domain and range?
Show Answer
Answer :
According to the given figure, $P=\{5,6,7\}, Q=\{3,4,5\}$
(i) $R=\{(x, y): y=x-2 ; x \in P\}$ or $R=\{(x, y): y=x-2$ for $x=5,6,7\}$
(ii) $R=\{(5,3),(6,4),(7,5)\}$
Domain of $R=\{5,6,7\}$
Range of $R=\{3,4,5\}$
5. Let $A=\{1,2,3,4,6\}$. Let $R$ be the relation on A defined by $\{(a, b): a, b \in A, b$ is exactly divisible by $a\}$.
Fig 2.7
(i) Write $R$ in roster form
(ii) Find the domain of $R$
(iii) Find the range of $R$.
Show Answer
Answer :
$A=\{1,2,3,4,6\}, R=\{(a, b): a, b \in A, b$ is exactly divisible by $a\}$
(i) $R=\{(1,1),(1,2),(1,3),(1,4),\\ (1,6),(2,2),(2,4),(2,6),(3,3),(3,6),(4,4),(6,6)\}$
(ii) Domain of $R=\{1,2,3,4,6\}$
(iii) Range of $R=\{1,2,3,4,6\}$
6. Determine the domain and range of the relation $R$ defined by $R=\{(x, x+5): x \in\{0,1,2,3,4,5\}\}$.
Show Answer
Answer :
$R=\{(x, x+5): x \in\{0,1,2,3,4,5\}\}$
$\therefore R=\{(0,5),(1,6),(2,7),(3,8),(4,9),(5,10)\}$
$\therefore$ Domain of $R=\{0,1,2,3,4,5\}$
Range of $R=\{5,6,7,8,9,10\}$
7. Write the relation $R=\{(x, x^{3}): x.$ is a prime number less than 10 $\}$ in roster form.
Show Answer
Answer :
$R=\{(x, x^{3}): x.$ is a prime number less than 10 $\}$
The prime numbers less than 10 are 2, 3, 5, and 7 .
$\therefore R=\{(2,8),(3,27),(5,125),(7,343) \}$
8. Let $A=\{x, y, z\}$ and $B=\{1,2\}$. Find the number of relations from $A$ to $B$.
Show Answer
Answer :
It is given that $A=\{x, y, z\}$ and $B=\{1,2\}$.
$\therefore A \times B=\{(x, 1),(x, 2),(y, 1),(y, 2),(z, 1),(z, 2)\}$
Since $n(A \times B)=6$, the number of subsets of $A \times B$ is $2^{6}$.
Therefore, the number of relations from $A$ to $B$ is $2^{6}$.
9. Let $R$ be the relation on $\mathbf{Z}$ defined by $R=\{(a, b): a, b \in \mathbf{Z}, a-b$ is an integer $\}$. Find the domain and range of $R$.
Show Answer
Answer :
$R=\{(a, b): a, b \in \mathbf{Z}, a-b$ is an integer $\}$
It is known that the difference between any two integers is always an integer.
$\therefore$ Domain of $R=Z$
Range of $R=Z$
