You need this lemma.
If f is surjective then by definition f[X]=Y.
So by the lemma we have at once .
Now if we need to show onto.
Suppose that f is not onto then .
But that means because .
Hello,
Can anyone help me with this question please ?
Let X and Y be sets, and let f : X → Y be a function. If A c ("Contained in") X, write f[A] for the set {f(x) | x ∈ A}. Prove that f is a surjection if and only if Y − f[A] c ("Contained in") f[X − A] for all A c ("Contained in") X.
Note: c means "contained in", I am sorry because I don't know how to type this symbol in and "c" is the closest representation to it.