Zeal Test Series 2019: Set Theory & Algebra - Functions

Question

Let f : A → B be function, where A = {1,2,3,4,5,6} and B = {1,2,3,4,5}.

If f(1) = 4 then how many surjective (onto) functions are possible ?

Comments

@Prince Sindhiya

type the questions instead of screenshots

https://gateoverflow.in/blog/5823/screenshot

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@shaik i access GO most of the time from mobile phone and i am not good with latex symbols even many times i tried when i was a new user but it took around 45 mins to 1 hour in typing the solutions to answering and all screenshot i posted are visible so i started Posting questions by taking the screenshot

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But from now onwards for simple question i will type the question

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for solutions no need, atleast for questions you can do that, hardly it take 5 min...

click on fx button, there you will type, no need to manually write the latex code.

searching By images is more difficult when we are finding duplicates...

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can you tell me that how to subscript and superscript ?

I didn't find that func button in this updated version

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check this

 

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Ohk fine :p

earlier it was bit easy because both function are very frequently use so sir make a separate  button for this two func

 btw thanks

Answer 1

related theory https://math.stackexchange.com/questions/334420/number-of-onto-functions

formula derivation is in given link

Comments

@gurdeep same solution is given for it

But i understood the second part that f(1) =4 that's why we have excluded that but why we are considering 4 in first part and why this 5Factorial is coming i wrote it's solution as 240 only please tell me the first part by explaining it a bit more

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 Prince Sindhiya

F(1)=4 is given and set b contain 5 element 

now in set A we have total 6 element including 1 

means 5 element excuding 1 

given that 5 element in B 

it means both set have 5 ,5 element where 1 is already related to 4  

now in the remaining 5 element of A we have to relate every element of A only one element of B here we can not relate 2 element of A to the single element of B because it we do so than this will not satisfy onto property because  some element will left unrelated in B 

so first element of A can relate to 5 ways 

and second element of A can relate to 4 ways

3 rd element 3 ways 

4th  element 2 ways

5th  element  1 ways  

so total ways is 5!

so in this process

 (one thing you should note in this process 4 element of B is related to two different element of A  )

 

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Gurdeep thnxx for such a beautifull solution

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Gurdeep Saini

tell me one thing

we exclude (1) from A is understandable

but why you exclude (4) from B ??

for every element y in the codomain B of f there is at least one element x in the domain A

one of the element from A {2,3,4,5,6} can  map to to B (4) right ?? 

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Prince Sindhiya  clear my doubt

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 Magma

element 1 of A already map to 4 of B  now we can ignore that elment 

and do the maping of rest 

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we can ignore 1

but why we ignore 4 also ??

because if 4 have 2 image in domain..then there's  no problem right ???

I know I'm wrong....but I want clarify my doubt..that's why I asking question

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because if 4 have 2 image in domain..then there's  no problem right ???

@magma this in included in first case

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what I wanted to say is that

this is possible right ??

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yes @magma this is possible 

but included in first case 

in the first case        1 element and one other of A related to 4 of B (shown by you in figure)

but in second case      only 1 element of A relate to the 4 element of B and rest elment of A relate to the remaining four element of B  

 

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thanks

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@Gurdeep Saini , great solution , so in this we have to take 2 cases, one with element 4 in the set B and one without element 4 in set B , because f(1)=4 , so element 4 from set B could be used or could not be used, so 2 cases.

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@shadan

yes

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@Gurdeep Saini

Good one :)

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i request to the person who downvoted this  answer 

please tell me what is wrong so that i dont repeat this mistake in exam 

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@Gurdeep Saini

who just down vote but didn't participate in the discussion, doesn't get this comment as notification

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@Shaik Masthan he/she should give the reason what is wrong in this answer 

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we can't do anything... it's his/her responsibility.... forget about it.. there is No problem with this solution