Relations and Functions

20. (i) Domain of the given relation = {2,5,8,11,14,17}

Since every element of the domain has one and only one image, the given relation is a fxⁿ.

So, domain = {2,5,8,11,14,17}

range = {1}

(ii) Domain of the given relation = {2,4,6,8,10,12,14}

Since every element of the domain has one and only one image, the given relation is a fxⁿ.

So, domain = {2, 4, 6, 8, 10, 12, 14}

range = {1,2,3,4,5,6,7}

(iii) Domain of the given relation = {1,2}

As element 1 has more than one image i.e., 3 and 5, the given relation is not a fxⁿ.

19. Given, R= { (a, b): a, b $\in$ z and a – b is an integer}

We know that, the difference of two integers is also an integer.

R= { (a, b): a – b $\in$ z & a, b $\in$ z}

Domain of R=Z.

Range of R= Z.

18. Given, A={x, y, z}so, n(A)=3

B={1,2} so n(B)=2

? n(A * B)=n(A) *n(B)=3 * 2=6

Hence, no. of relation from A to B=Number of subsets of A * B

=2⁶

=64.

17. GivenR= { (x, x³) : x is a prime number less than 10}

R = { (x, x³) : x = 2,3,5,7}

= { (2,2³), (3,3³), (5,5³), (7,7³)}

= { (2,8), (3,27), (5,125), (7,343)}

16. Given, R = { (x, x+5): x $\in$  {0,1,2,3,4,5}

= { (0,0+5), (1,1+5), (2,2+5), (3,3+5), (4,4+5), (5,5+5)}

= { (0,5), (1,6), (2,7), (3,8), (4,9), (5,10)}

So, domain of R= {0,1,2,3,4,5}

range of R= {5,6,7,8,9,10}

15. Given, 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}

14. As R is a relation from set P to Q.

(i) R = { (x, y): x – 2 = y ; 5 ≤ x ≤ 7}

(ii) R = { (5,3), (6,4), (7,5)}

Domain of R= {5,6,7}

range of R= {3,4,5}

13. Given, A= {1,2,3,5}

B= {4,6,9}

R= { (x, y) : the difference of x & y is odd; x $\in$ A, y $\in$ B}.

= { (x, y):|x – y| is odd and x $\in$ A, y $\in$ B}

= { (1,4), (1,6), (2,9), (3,4), (3,6), (5,4), (5,6)}.

12. Given, R = { (x, y): y = x + 5, x is a natural number less than 4; x, y $\in$ N}

= { (x, y): y = x + 5; x, y $\in$ N and x < 4}.

= { (1,1+5), (2,2+5), (3,3+5)}

= { (1,6), (2,7), (3,8)}

So, domain of R = {1,2,3}

range of R = {6,7,8}

11. Given, A = {1,2,3, …, 14}

R = { (x, y): 3x – y = 0; x, y $\in$ A}

= { (x, y): 3x = y; x, y $\in$ A}.

= { (1,3), (2,6), (3,9), (4,12)}

Domain of R is the set of all the first elements of the ordered pairs in R

So, domain of R= {1,2,3,4}

Codomain of R is the whole set A.

So, codomain of R= {1,2,3, …, 14}

Range of R is the set of all the second elements of the ordered pains in R.

So, range of R= {3,6,9,12}