Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.

 ncert cbse 11th chapter 2 relations and functions miscellaneous exercise

 

8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

 

(1,1)  is in f 

 

so

x=1, y=1 or f(1)=1 means 

f(1) = a(1) + b can be rewritten as 1= a + b  or a+b=1

 

(2,3) is in f

so 

x=2 , y = 3 or f(2)  = 3 

means

f(2) = a(2) + b can be re written as 3 = 2a+b or 2a+b =3

  a+b=1

2a+b =3

solve by elimination method

subtracting gives 

a = 2 

resubstitute to get b = (-1) 

9. Let R be a relation from N to N defined by

 R = {(a, b) : a, b ∈ N and a = (b^2) }. 

Are the following true?
(i) (a,a) ∈ R, for all a ∈ N
(ii) (a,b) ∈ R, implies (b,a) ∈ R
(iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R.

 

(i) if b =2, 

a=(b^2) =(2^2) = 4

so only (4,2)∈ R and (2,2) is not in R [ because 2^2 is not 2 ]

OR assume  (2,2)∈ R

a=2 , b=2 so that a=(b^2)  is true or 2 =(2^2) which is clearly false

so (2,2) is not in R

 

(a,a) ∈ R, for all a ∈ N is FALSE.

(ii)

if b =2, 

a=(b^2) =(2^2) = 4

so (4,2)∈ R

but if b=4 ,then a =b^2 = 4^2 =16

so that only (16,4) ∈ R and (2,4) is not in R [ because 4^2 is not 2 ]

 (a,b) ∈ R, implies (b,a) ∈ R is FALSE.

(iii)

if b=4 ,then a =b^2 = 4^2 =16

so that  (16,4) ∈ R

if b =2, 

a=(b^2) =(2^2) = 4

so (4,2)∈ R

now both

 (16,4) ∈ R and (4,2)∈ R

but (16,2)  is not in R because (2^2) is not 16

 (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R is FALSE

 

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11th cbse ncert chapter 2 relations and functions miscellaneous exercise

 

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to  Z? Justify your answer.

solution

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.

solution 

 8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

solution