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.
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Answer :
The relation $R$ from $A$ to $A$ is given as
$ R=\left{(x, y): \ \ 3^x-y=0 ; \ \ x, y \in A\right} $
i.e., $ \ R={(x, y): \ \ 3 x=y ; \ \ 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 co-domain of the relation $R$
$\therefore \ \ $ Codomain of $\mathrm{R}=\mathrm{A}={1,2,3, \ldots ., 14}$
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.
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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.
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Answer :
$A={1,2,3,5}$ and $B={4,6,9}$
$R={(x, y)$ : the difference between $x$ and y is 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 Fig $2.7$ 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?
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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 $a}$.
(i) Write $R$ in roster form
(ii) Find the domain of $R$
(iii) Find the range of $R$
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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 $R={(x, x+5): x \in{0,1,2,3,4,5}}$.
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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.
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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 $B$.
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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 $R$.
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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 $R=Z$
BETA
