zooxanthellae said:Your basic idea is correct. However, I would be careful with how exactly you write this out. Your first statement, "Since f is onto then there exists an x in X such that f(x) = y, for all y in Y" is not quite correct. The way you have it arranged, it reads as if there is one specific x in X that maps to all values of y in Y.
Of course this is not what you mean, but if you're starting out with proofs (which I'm assuming you are) then it is good to be precise at first. "For a given y in Y, there exists some x in X such that f(x) = y."
Hope that helps!
Being "onto" means that every element in the range of the function is mapped to by at least one element in the domain. In other words, every output has at least one corresponding input.
To prove that g°f is onto, you need to show that for every element in the range of g, there exists at least one element in the domain of f that maps to it. This can be done by using the fact that g is onto and f is onto, and showing that the combination of the two functions still covers the entire range of g.
Being "one-to-one" means that each element in the domain is mapped to a unique element in the range. This is different from being "onto" because in an onto function, multiple elements in the domain can map to the same element in the range.
Yes, a function can be onto but not one-to-one. This means that there are multiple elements in the domain that map to the same element in the range. However, every element in the range is still covered by at least one element in the domain.
Proving g°f is onto is important because it ensures that every element in the range of the combined function is covered by at least one element in the domain. This can help validate the results of the function and ensure that it is a valid and useful tool in solving mathematical problems.