GATE CSE 1996 | Question: 1.3

Question

Suppose $X$ and $Y$ are sets and $|X| \text{ and } |Y|$ are their respective cardinality. It is given that there are exactly $97$ functions from $X$ to $Y$. From this one can conclude that

• A. $|X| =1, |Y| =97$
• B. $|X| =97, |Y| =1$
• C. $|X| =97, |Y| =97$
• D. None of the above

Comments

1. 97 functions.
2. 1 function.
3. 97^97 functions.

Answer 1

We can say $|Y|^{|X|} = 97.$ Only option A satisfies this. Still, this can be concluded only because $97$ is a prime number and hence no other power gives $97.$

Comments

@Mohitdas , yes we can't have one many function, but here we have 97 “different functions", each function corresponding to one element of X and one of 97 elements.

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Sir function is a relation for each input it has one output

Here 1 input has 97 different outputs

And I'm not getting “different functions”

Y and X are 2 sets

Can you please draw so that I can visualize..

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Here

Answer 2

I think this question is not right and I have not got the right explanation of this question anywhere.

Everywhere is concluding $|Y|^{|X|} = 97^1 $ only satisfying but if it is satisfying that

Number of functions means how many mappings possible from domain to codomain

So from above 5 can be mapped by 97 ways so,

option (A)

Comments

hii @pritishc please read my answer once as well as selected question's discussion once...

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@Rishi yadav you should read it again. There's not even any ambiguity here. Probably you're not getting the definition of function properly. Please refer a standard book and it should be clear.

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My bad sir i got my mistake

Answer 3

How to know whether 97 is a prime or not ? 

Take sqrt of 97 = 10 (approx)

now check whether 97 is divisible by all prime numbers which are <= 10 

If no then it is a prime number ... 97 is not divisible by 2,3,5,7 .. so 97 is a prime number...

So 97 cannot be written as a product of 2 numbers(apart from 97*1)...And so 97 cannot be represented as n^m , (apart from 97¹ ) ...

So only way we could have got 97 functions is using 97¹ which is possible only if 1 element is in domain and 97 elements in co-domain.

B) says number of fuinctions is 1.

C) says number of functions is 97⁹⁷. 

Comments

Please explain that how option a is even a function?

1 element in X connected to all (97) elements in Y ....i think it does not satisfy the function definition..

In this case shouldn't be the answer d

Answer 4

 functions = $co-domaindomain^{domain}$ and number of function given 97 so this is possible only if co-domain = 97 and domain = 1

 

Comments

we can't map one to many relationship in function . here we have 97 choices to select 1 function from "domain to co-domain"

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Thanks