Prove that a function is surjective but not bijective

[mandlao]

I'm trying to prove that the function f:P(ℤ)→P(ℕ) defined by f(X)=X∩ℕ is surjective but not bijective.
In order to do this, I need to prove that f is surjective and not injective. To prove that it is surjective, I have to suppose B∈P(ℕ) to prove that there exists A∈P(ℤ) for which f(A)=f(B). Also, it might be easier to prove that the image of f is P(ℕ), but I don't know how to do this (I'm doing it by double inclusion but I can't seem to prove that P(ℕ)⊂X∩ℕ .
Thanks.

 

[pka]

I'm trying to prove that the function f:P(ℤ)→P(ℕ) defined by f(X)=X∩ℕ is surjective but not bijective.
In order to do this, I need to prove that f is surjective and not injective. To prove that it is surjective, I have to suppose B∈P(ℕ) to prove that there exists A∈P(ℤ) for which f(A)=f(B). Also, it might be easier to prove that the image of f is P(ℕ), but I don't know how to do this (I'm doing it by double inclusion but I can't seem to prove that P(ℕ)⊂X∩ℕ .

Clearly the function is surjective. Can you prove that?

If \(\displaystyle S=(-\infty,0)\cap\mathbb{Z}~\&~T=(-\infty,-2)\cap\mathbb{Z}\) is it true that \(\displaystyle f(S)=f(T)~?\)

 

[mandlao]

Clearly the function is surjective. Can you prove that?

If \(\displaystyle S=(-\infty,0)\cap\mathbb{Z}~\&~T=(-\infty,-2)\cap\mathbb{Z}\) is it true that \(\displaystyle f(S)=f(T)~?\)

No, they map different sets. I'm confused, how can I prove that it is surjective?

 

[pka]

how can I prove that it is surjective?

If \(\displaystyle K\in\mathcal{P}(\mathbb{N})~\&~X=K\cap\mathbb{Z}\) then \(\displaystyle f(X)=~?\)

 

[mandlao]

If \(\displaystyle K\in\mathcal{P}(\mathbb{N})~\&~X=K\cap\mathbb{Z}\) then \(\displaystyle f(X)=~?\)

So f(X) = K. That's how you prove surjectivity in this case? (the ∩ doesn't "invert"?)

 

[pka]

So f(X) = K. That's how you prove surjectivity in this case? (the ∩ doesn't "invert"?)

Do you even know the definition of surjective?

If so, what is the definition and how is it applied in this problem?