Question

How many onto functions are there from a set with six elements to a set with four elements?m

Answer 1

There are 2160 onto functions from a set with six elements to a set with four elements. This calculation is based on the Stirling number of the second kind formula.

An onto function (also known as a surjective function) is a function in which every element in the codomain is mapped to by at least one element in the domain.

Let's consider a set with six elements as the domain (A) and a set with four elements as the codomain (B).

Since the function must be onto, each element in set B must have at least one pre-image in set A. The number of onto functions from a set with m elements to a set with n elements is given by:

`\[ n! \left[{{m \brace n}}\right] \]`

where
`\[{{m \brace n}}\]` denotes the Stirling number of the second kind.

For this case (m=6, n=4), the number of onto functions is:

`\[ 4! \left[{{6 \brace 4}}\right] = 24 * 90 = 2160 \]`

Therefore, there are 2160 onto functions from a set with six elements to a set with four elements.