NCERT Solutions for Class 11 Maths Chapter 2 Exercise 2.2 - Relations and Functions

Question 1: Let A = {1, 2, 3,...,14}. Define a relation R from A to A by $R=\{(x,y):3x-y=0,wherex,yϵA\}$ .Write down its domain, codomain and range.

Answer:

It is given that
$A=\{1,2,3,...,14\}andR=\{(x,y):3x-y=0,wherex,yϵA\}$
Now, the relation R from A to A is given as
$R=\{(x,y):3x-y=0,wherex,yϵA\}$
Therefore,
The relation in roaster form is , $,R=\{(1,3),(2,6),(3,9),(4,12)\}$
Now,
We know that the Domain of R = set of all first elements of the order pairs in the relation
Therefore,
Domain of $R=\{1,2,3,4\}$
And
Codomain of R = the whole set A
i.e. Codomain of $R=\{1,2,3,...,14\}$
Now,
Range of R = set of all second elements of the order pairs in the relation.
Therefore,
range of $R=\{3,6,9,12\}$

Question 2: Define a relation R on the set N of natural numbers by $R=\{(x,y):y=x+5,x\}$ is a natural number less than $4;x,yϵN\}$. Depict this relationship using roster form. Write down the domain and the range.

Answer:

As x is a natural number which is less than 4.
Therefore,
the relation in roaster form is, $R=\{(1,6),(2,7),(3,8)\}$
As Domain of R = set of all first elements of the order pairs in the relation.
Therefore,
Domain of $R=\{1,2,3\}$

Now,
Range of R = set of all second elements of the order pairs in the relation.
Therefore,
the range of $R=\{6,7,8\}$

Therefore, domain and the range are $\{1,2,3\}and\{6,7,8\}$ respectively

Question 3: A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by $R=\{(x,y):thediffrencebetweenxandyisodd;xϵA,yϵB\}$. Write R in roster form.

Answer:

It is given that
A = {1, 2, 3, 5} and B = {4, 6, 9}
And
$R=\{(x,y):thediffrencebetweenxandyisodd;xϵA,yϵB\}$
Now, it is given that the difference should be odd. Let us take all possible differences.
(1 - 4) = - 3, (1 - 6) = - 5, (1 - 9) = - 8(2 - 4) = - 2, (2 - 6) = - 4, (2 - 9) = - 7(3 - 4) = - 1, (3 - 6) = - 3, (3 - 9) = - 6(5 - 4) = 1, (5 - 6) = - 1, (5 - 9) = - 4
Taking the difference which are odd we get,

Therefore,
the relation in roaster form, $R=\{(1,4),(1,6),(2,9),(3,4),(3,6),(5,4),(5,6)\}$

Question 4: (i) The Fig2.7 shows a relationship between the sets P and Q. Write this relation in set-builder form

Answer:

It is given in the figure that

P = {5,6,7}, Q = {3,4,5}

Therefore,
the relation in set builder form is ,
$R=\left\{(x,y):y=x-2;xϵP\right\}$
OR
$R=\left\{(x,y):y=x-2;forx=5,6,7\right\}$

Question 4: (ii) The Fig2.7 shows a relationship between the sets P and Q. Write this relation roster form. What is it domain and range?

Answer:

From the given figure. we observe that

P = {5,6,7}, Q = {3,4,5}

And the relation in roaster form is , $R=\left\{(5,3),(6,4),(7,5)\right\}$

As Domain of R = set of all first elements of the order pairs in the relation.
Therefore,
Domain of $R=\left\{5,6,7\right\}$

Now,
Range of R = set of all second elements of the order pairs in the relation.
Therefore,
the range of $R=\left\{3,4,5\right\}$

Question 5: (i) Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by $\{(a,b):a,bϵA,bisexactlydivisiblebya\}$ Write R in roster form

Answer:

It is given that
A = {1, 2, 3, 4, 6}
And
$R=\{(a,b):a,bϵA,bisexactlydivisiblebya\}$

Therefore,
the relation in roaster form is , $R=\left\{(1,1),(1,2),(1,3),(1,4),(1,6),(2,2),(2,4),(2,6),(3,3),(3,6),(4,4),(6,6)\right\}$

Question 5: (ii) Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by $\{(a,b):a,bϵA,bisexactlydivisiblebya\}$ Find the domain of R

Answer:

It is given that
A = {1, 2, 3, 4, 6}
And
$R=\{(a,b):a,bϵA,bisexactlydivisiblebya\}$
Now,
As Domain of R = set of all first elements of the order pairs in the relation.
Therefore,
Domain of $R=\left\{1,2,3,4,6\right\}$

Question 5: (iii) Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by $\{(a,b):a,bϵA,bisexactlydivisiblebya\}$ Find the range of R.

Answer:

It is given that
A = {1, 2, 3, 4, 6}
And
$R=\{(a,b):a,bϵA,bisexactlydivisiblebya\}$
Now,
As the range of R = set of all second elements of the order pairs in the relation.
Therefore,
Range of $R=\left\{1,2,3,4,6\right\}$

Question 6: Determine the domain and range of the relation R defined by $R=\{(x,x+5):xϵ\{0,1,2,3,,4,5\}\}$

Answer:

It is given that
$R=\{(x,x+5):xϵ\{0,1,2,3,,4,5\}\}$

Therefore,
the relation in roaster form is , $R=\left\{(0,5),(1,6),(2,7),(3,8),(4,9),(5,10)\right\}$

Now,
As Domain of R = set of all first elements of the order pairs in the relation.
Therefore,
Domain of $R=\left\{0,1,2,3,4,5\right\}$

Now,
As Range of R = set of all second elements of the order pairs in the relation.
Range of $R=\left\{5,6,7,8,9,10\right\}$

Therefore, the domain and range of the relation R is $\{0,1,2,3,4,5\}and\left\{5,6,7,8,9,10\right\}$ respectively

Question 7: Write the relation $R=\{(x,x^3):xisaprimenumberlessthan10\}$ in roster form.

Answer:

It is given that
$R=\{(x,x^3):xisaprimenumberlessthan10\}$
Now,
As we know the prime number less than 10 are 2, 3, 5 and 7.
Therefore,
the relation in roaster form is , $R=\left\{(2,8),(3,27),(5,125),(7,343)\right\}$

Question 8: Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.

Answer:

It is given that
A = {x, y, z} and B = {1, 2}
Now,
$A\times B=\left\{(x,1),(x,2),(y,1),(y,2),(z,1),(z,2)\right\}$
Therefore,
$n(A\times B)=6$
Then, the number of subsets of the set $(A\times B)=2^n=2^6$

Therefore, the number of relations from A to B is $2^6$

Question 9:Let R be the relation on Z defined by $R=\{(a,b):a,bϵZ,a-bisaninteger\}$
Find the domain and range of R.

Answer:

It is given that
$R=\{(a,b):a,bϵZ,a-bisaninteger\}$
Now, as we know that the difference between any two integers is always an integer.
And
As Domain of R = set of all first elements of the order pairs in the relation.
Therefore,
The domain of R = Z

Now,
Range of R = set of all second elements of the order pairs in the relation.
Therefore,
range of R = Z

Therefore, the domain and range of R is Z and Z respectively