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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.
in Set Theory & Algebra by Boss (30.8k points) | 632 views
0

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

3 Answers

+21 votes
Best answer

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 $n^m .$

by Loyal (8k points)
edited by
+1
You are absolutely correct with the definition of single valued function but here they are asking about no of functions .

You left out many cases for instance f(1)=f(2)=a,f(3)=b...etc these all instances should be considered !
+1
The solution is corrected now.Thanks, bro.
+8 votes

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

so i tink nare the total number of the Single valued function are there. 

by Active (3.8k points)
0

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

0 votes
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.
by (367 points)
0
@jpranvc why so its not making sense can you elaborate what are you thinking.?
0
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?
0
in gate everything specified you cant assume anything...so thats why qsn make sense...you are correct bdw..!
0
Thanks, @Shubham
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