if X={1,....,n} and Y={1,2,3}

how many functions are there from X to Y with image containing exactly 1 element?

- Dec 21st 2012, 08:52 AMdarren86How many functions from X to Y with image containing 1 element?
if X={1,....,n} and Y={1,2,3}

how many functions are there from X to Y with image containing exactly 1 element? - Dec 21st 2012, 09:09 AMPlatoRe: How many functions from X to Y with image containing 1 element?
- Dec 21st 2012, 10:33 AMdarren86Re: How many functions from X to Y with image containing 1 element?
- Dec 21st 2012, 11:06 AMPlatoRe: How many functions from X to Y with image containing 1 element?
- Dec 21st 2012, 03:13 PMDevenoRe: How many functions from X to Y with image containing 1 element?
in general, the number of possible functions from a set X to a set Y is:

|Y|^{|X|}.

if Y is but a single element, then this number is 1 (all functions are the same: f(x) = y, the single element of Y for every x).

in this case, we have 3 single-element subsets of Y, {1},{2} and {3}.

there is only one function f:X-->{1}

one function f:X-->{2}

one function f:X-->{3}

(namely, the constant function in each case).

there are 2^{n}functions if the image is a 2-element subset of Y, and 3^{n}functions if the image is a 3-element subset. convince yourself that this is so for small values of n like 2 or 3. - Dec 22nd 2012, 11:48 AMdarren86Re: How many functions from X to Y with image containing 1 element?
- Dec 22nd 2012, 12:41 PMPlatoRe: How many functions from X to Y with image containing 1 element?
Do you understand what a function is?[/TEX]

If $\displaystyle A=\{1,2,3,4,5\}~\&~b=\{a,b,c\}$.

Then any function $\displaystyle A\to B$ looks like:

$\displaystyle \{(1,\underline{~~~}),~(2,\underline{~~~}), (3, \underline{~~~}),(4,\underline{~~~}),(5,\underline {~~~})\}$

Now there are three choices to go into each blank.

So there are $\displaystyle 3^5$ functions from $\displaystyle A\to B$.

The number of functions from $\displaystyle A\to B$ is $\displaystyle |B|^{|A|}$ - Dec 23rd 2012, 02:30 AMDevenoRe: How many functions from X to Y with image containing 1 element?
i don't think you understand what i meant:

there are 2^{n}functions from X = {1,2...,n} to Y = {1,3}

2^{n}MORE functions from X to Y = {2,3}

2^{n}MORE functions from X to Y = {1,2}

there is ONE function from X to {1}, one more from X to {2}, and one more from X to {3} (all of these are constant functions which have been counted as one of the 3*2^{n}above)

there is also a unique function from X to the empty set: the "empty function" (think of a blank sheet of paper you were going to draw the graph of f on, but never did).

there are 3^{n}functions from X to Y = {1,2,3} some of these are the functions above i've already listed.