# Thread: How many functions from X to Y with image containing 1 element?

1. ## How 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?

2. ## Re: How many functions from X to Y with image containing 1 element?

Originally Posted by darren86
if X={1,....,n} and Y={1,2,3}, how many functions are there from X to Y with image containing exactly 1 element?

As stated the answer is clearly three.
But I think that is not what is meant?

3. ## Re: How many functions from X to Y with image containing 1 element?

Originally Posted by Plato
As stated the answer is clearly three.
But I think that is not what is meant?
That is word for word the question. Would you mind explaining a little why its 3? I don't really understand what the question is asking. What does it mean that the image contains exactly one element, etc?

Thanks

4. ## Re: How many functions from X to Y with image containing 1 element?

Originally Posted by darren86
That is word for word the question. Would you mind explaining a little why its 3? I don't really understand what the question is asking. What does it mean that the image contains exactly one element.

If the image has exactly one element then
The function look like one of these:
{(1,1),(2,1),...,(n,1)}, {(1,2),(2,2),...,(n,2)}, or {(1,3),(2,3),...,(n,3)}.

That is only three possible functions.

5. ## Re: 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 2n functions if the image is a 2-element subset of Y, and 3n functions if the image is a 3-element subset. convince yourself that this is so for small values of n like 2 or 3.

6. ## Re: How many functions from X to Y with image containing 1 element?

Originally Posted by Deveno
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 2n functions if the image is a 2-element subset of Y, and 3n functions if the image is a 3-element subset. convince yourself that this is so for small values of n like 2 or 3.
there are 3, two element subset of Y, so how can there be 2^n functions?

7. ## Re: How many functions from X to Y with image containing 1 element?

Originally Posted by darren86
there are 3, two element subset of Y, so how can there be 2^n functions?
Do you understand what a function is?[/TEX]

If $A=\{1,2,3,4,5\}~\&~b=\{a,b,c\}$.
Then any function $A\to B$ looks like:
$\{(1,\underline{~~~}),~(2,\underline{~~~}), (3, \underline{~~~}),(4,\underline{~~~}),(5,\underline {~~~})\}$

Now there are three choices to go into each blank.
So there are $3^5$ functions from $A\to B$.

The number of functions from $A\to B$ is $|B|^{|A|}$

8. ## Re: How many functions from X to Y with image containing 1 element?

Originally Posted by darren86
there are 3, two element subset of Y, so how can there be 2^n functions?
i don't think you understand what i meant:

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

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

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