Mock Test Math Q. 13

[Rajesh Patel]

I am unable to understan why it is not surjective

please explain

[Mathematics 1 Support 2]

Dear learner,

Kindly go through the mock test solution and let us know what exactly you are not able to understand. It will be good if you can post your working here.

Thanks and regards 

Vicky Kumar Sharma 

Maths-1 instructor

IITM Online Degree 

[Rajesh Patel]

You said 3 does not have pre image

how ?

i am putting 0,1,2,3,4,5,6 

and getting value like 1,0,1,9,44....

the preimage thing i didn't get

[Aditi Prasad]

 a function is said to be surjective or onto, only if whatever value you put in n, and anywhere you are getting back the same value on the f(n) side, i.e  for every element y in the codomain Y of f, there is at least one element x in the domain X of f.
you rightly said, "i am putting 0,1,2,3,4,5,6 

and getting value like 1,0,1,9,44...."

and you can see however long you take it, you will not get 3 as output on f(n) side. The values you put such as "0,1,2,3,4,5..." is known as codomain and the output values is the range.
for a function to be surjective codomain must be equal to the range.

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[Rajesh Patel]

Thank you so much for your reply,

but i am getting more confused here Sorry

can you please tell me what is the domain , Range and codomain here

On Friday, November 20, 2020 at 12:25:00 PM UTC+5:30 Rajesh Patel wrote:

THank you so much f

[Malabika Guha Mustafi]

Let us define a function f:N→N such that f(0)=1,f(1)=0, and f(n)=(n−1)[f(n−1)+f(n−2)] for all n≥2. Define the following sets,

• S1={n|f(n) is even, n≤6}
• S2={n|f(n) is odd, n≤6}
• T={n|f(n) is prime }
  
  

So, this says that the function "f" has a domain of "N" (the natural numbers), and a codomain of "N" also.

f(0)=1

f(1)=0

f(2)= 1[f(1) + f(0)]=1[0+1]= 1

so , it is not Injective.

f(3) =2[ f(2)+f(1)]=2

f(4)= 3[f(3) +f(2)]=9...so on

clearly, the range means actual output is not taking all natural numbers as it's codomain is defined. A function is called as surjective when codomain = range.

so, this function neither injective or subjective.

The function is a quadratic one so it can not  take all N values as range so it is never be surjective. 

Hope, it clears your doubt.

[Aditi Prasad]

domain: what can go into a function; codomain: what may possibly come out of a function; range: what actually comes out of a function. 

[Rajesh Patel]

Thanks for your response

Now it's clear