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.

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The Correct Answer (|X| =1 AND |Y|= 97)

Imagine you have a machine with only one button (Set $X$), but the machine can output 97 different colors (Set $Y$).

• Set $X$: $\{\text{Button}\}$

• Set $Y$: $\{\text{Red, Blue, Green... up to 97 colors}\}$

How many different ways can you program this machine?

1. Program it so Button $\rightarrow$ Red (Function 1)

2. Program it so Button $\rightarrow$ Blue (Function 2)

3. Program it so Button $\rightarrow$ Green (Function 3)

   ...and so on, up to the 97th color.

Because you only have one decision to make, and 97 options to choose from, there are exactly 97 possible functions. Mathematically: $97^1 = 97$.

Scenario B: The Trap Answer (|X|=97 AND |Y|=1)

Now flip the machine. Imagine a giant control panel with 97 different buttons (Set $X$), but the machine only has one possible output color, let's say "Black" (Set $Y$).

• Set $X$: $\{\text{Button 1, Button 2, ... Button 97}\}$

• Set $Y$: $\{\text{Black}\}$

How many different ways can you program this machine?

Let's make our choices:

• Where does Button 1 go? It must go to Black.

• Where does Button 2 go? It must go to Black.

• Where does Button 97 go? It must go to Black.

Every single arrow from the 97 inputs converges onto the exact same single output. Because there are zero alternative choices being made, this entire setup represents just 1 single, massive function. Mathematically: $1^{97} = 1$.

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

@Pranay_VG yes it is function

as you can see below set A has two elements and set B has two elements , i have taken only two elements in both domain and codomain is understand simply.

for codmain (B) = 2 and domain (A) = 2
each domain has 2 choices { 2 * 2 } 
so 2^(2) = 4 functions are presents , now below are the four function 

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now is it function?

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No @Pranay_VG ,

because , in your image one domain element have multiple images in codomain! 

for it to be function every domain element should have exactly one image in codomain

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

@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

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exactly 97 functions is keyword here ... means
Since \(\vert{}X\vert{} = 1\)... let's say the set is just \(X = \{x_1\}\)...Since \(\vert{}Y\vert{} = 97\), let's say the set is \(Y = \{y_1, y_2, y_3, \dots, y_{97}\}\)...To create a function, you must choose
where \(x_{1}\) goes.
\(f_{1}\) sends \(x_1 \rightarrow y_1\)
\(f_{2}\) sends \(x_1 \rightarrow y_2\)
\(\dots \)
\(f_{97}\) sends \(x_1 \rightarrow y_{97}\)
 

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

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@gari I think you are talking about cross product or relations.

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

Answer 5

.

Answer 6

let |X|=n, |Y|=m here we apply, $m^{n}$=97, i.e, $|Y|^{|X|}$=97
97 is a prime number therefore it only makes sense that |Y|=97 and |X|=1.
Also this is not one to many mapping.
Here domain only contains 1 element and co domain has 97 elements.
Therefore there are 97 possible functions(or 97 possible mappings) from X to Y.