I have already done the proof, but I just want to make sure it is correct!
f: ZxZ -> Z f(m,n)=m+n
Prove that f is onto using the definition of onto.
Suppose that f(m,n)=m+n.
Then, f(m,n)=a iff m+n=a iff m=a-n which is in Z.
Therefore, f(m+n)=m+n is onto.
Is this correct? Or proper?
you can think of a function as something which starts at a source "a", and "hits a target b":
the only restriction is that each source can only pick "one target".
normally, the unique target of a, is called f(a), so we can write: f:a→f(a).
a function is onto if: "every target gets hit". what that means is: given any target b, we have to find at least one source a with f:a→b, that is at least one a with f(a) = b, for every b.
in YOUR function, the targets live in the set of integers. so to prove that f is onto, we need to find a pair (ANY pair) that adds to a given integer k, and we have to do this for EACH integer k.
well, one such pair is (k,0)...another is (0,k). for both of these pairs, it is straightforward to verify that:
f(k,0) = k+0 = k
f(0,k) = 0+k = k.
but, pick your favorite integer m (it doesn't matter which one you pick, just pick one, and stick with it).
then the pair (m,k-m) will work just as well:
f(m,k-m) = m+(k-m) = (m+k)-m = (k+m)-m = k+(m-m) = k+0 = k. the thing to be clear about, is that while we are doing this for each k, it is conceptually clearer, if we use the same "m", over and over. Plato chose m = 0, from which i deduce that 0 is his favorite integer.