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