GATE CSE 1989 | Question: 13c

Question

Find the number of single valued functions from set $A$ to another set $B,$ given that the cardinalities of the sets $A$ and $B$ are $m$ and $n$ respectively.

Comments

single valued functions from A to B  = number of functions from A to B

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This is a multi-valued function

 

This is a single valued function

Answer 1

A single-valued function is an emphatic term for a mathematical function in the usual sense. That is, each element of the function's domain maps to a single, well-defined element of its range. By default, we always consider function as a single-valued function except when clearly mentioned that function is a multi-valued function. 

So, the number of single valued functions from $A$ to $B =$  number of functions from $A$ to $B$.

Lets take an example:

$A =\{1,2\} ,B=\{a,b \}$

1. $f(1)=f(2)=a$
2. $f(1)=f(2)=b$
3. $f(1)=a$ and $f(2) =b$
4. $f(1)=b$ and $f(2) =a$

The total number of single-valued functions from set $A$ to another set  $B =∣ B∣^{∣A ∣}​​​=n^m$. This is because for every element in $A$ we have $\mid B\mid$ possibilities in the function. 

The correct answer is $n^m .$

Comments

@Deepak Poonia Sir a single valued function should be injective because every element of the domain maps to exactly one element in codomain  which is the definition for one-one functions

So should’nt the answer be nPm??

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and @Deepak Poonia Sir what are multi valued functions??

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(Single Valued Function) = Function

Multi-valued function is not universally defined, So, if they ask, they will give definition of it.

https://en.wikipedia.org/wiki/Multivalued_function

Answer 2

single valued are the function which has the domain single element map to only one element in Range.

so i tink n^m are the total number of the Single valued function are there. 

Comments

Yeah, this is the correct definition of a Single valued function.Thanks :-)

Answer 3

As per functions definition in general, "all elements of set A should be mapped to some element in set B, no element from set A should be mapped to more than one value in set B."

Then this question is not making any sense. Please correct me if I am making any mistake.

Comments

@jpranvc why so its not making sense can you elaborate what are you thinking.?

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Hello shubham ,

Thanks for the reply, as we can answer directly using the definition of function I think mentioned in earlier post. Is it correct?

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in gate everything specified you cant assume anything...so thats why qsn make sense...you are correct bdw..!

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Thanks, @Shubham

Answer 4

every element of A has $n$ choices in B

i.e. $n$ choices for 1st element in A and 

     $n$ choices for 2nd element in A and

    $n$ choices for 3rd element in A and

    ……………………………….

   …………………………………

   $n$ choices for $m^{th}$ element in A

By product rule,

no of functions from A to B = $n*n*n*……..(m times)$=$n^m$