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

18 votes

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.

23 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:

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

- $f(1)=f(2)=a$
- $f(1)=f(2)=b$
- $f(1)=a$ and $f(2) =b$
- $f(1)=b$ 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 $n^m .$

8 votes

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

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.

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