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Math Help - If m<n, find # of onto functions from A onto B

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    If m<n, find # of onto functions from A onto B

    We have set A with m elements and set B with n elements. If m<n, Find the number of functions from A onto B.


    Okay so a Function is A-->B is a relation f:A-->B then 1.) Domf=A, 2.) Every x in DomA has unique image in B. (if (x,y) in f and (x,z) in f then y=z)

    And onto is if Rngf=B (every image in B has a preimage in A)


    Now i did an example with A={a,b} and B={c,d,e} so m<n.

    i used the string approach

    Set B: c d e
    Choices from set A: 2 1 0

    Apply the product rules is 1*2*0=0 so we have 0 onto fns from A to B

    Now to prove for all numbers.

    set A={a(1), a(2)....a(m)}
    set B={a(1),a(2)...a(m)...a(n)}

    So m<n

    so with strings

    set B: a(1) a(2)... a(m)...(an)
    # choices from set A: a(n) a(n-1)..a(m)....0

    so applying the product rule a(n)*a(n-1)*...*a(m)*...0=0

    so if m<n then there can be 0 onto functions.


    Is this wrong? My teacher says its more complicated but i think hes being complicated. Please any feedback would be gracious.


    My teacher told me this:
    This problems is more involved than that. The easiest way to solve it is by counting the number of non-onto functions (missing 1, 2, 3, ...n target elements) and then conclude that # onto functions=#functions-# non-onto functions. Get a feeling for this by working with |A|=5, |B|=4: 1. Count the number of functions from A to B 2. Count the number of non-onto functions from A to B Number of functions missing one target + # functions missing two targets +# functions missing 3 targets + # functions missing 4 targets 3. Now count # onto functions
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    Re: If m<n, find # of onto functions from A onto B

    Quote Originally Posted by Aquameatwad View Post
    We have set A with m elements and set B with n elements. If m<n, Find the number of functions from A onto B.
    If |A|=m<n=|B| there are no onto functions from A\to B

    Quote Originally Posted by Aquameatwad View Post
    We have set A with m elements and set B with n elements. Find the number of functions from A onto B.
    For finite sets:
    On the other hand, if |A|\ge |B|
    then there are onto functions A\to B.

    Say |A|=M,~|B|=N~\&~M\ge N the using the inclusion/exclusion counting rule we can find \sum\limits_{k = 0}^N {(-1)^k\binom{N}{k}(N-k)^M
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    Re: If m<n, find # of onto functions from A onto B

    Quote Originally Posted by Plato View Post
    If |A|=m<n=|B| there are no onto functions from A\to B
    Yep. This is even true for infinite sets, since |X| < |Y| ::= "There is an injection from X to Y but no bijection." The only way that can happen is if there is no possible surjection from Y to X.

    (Notably, the converse does not hold in absence of choice: a surjection from X to Y does not necessarily imply a injection exists from Y to X, at least not in ZF~C.)
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    Re: If m<n, find # of onto functions from A onto B

    My string approach didn't line up the way i typed it. But i think i am right in showing there are 0 onto functions

    i think my teacher was mistaken then. he must have been thinking m>n.
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