Understanding Sets & Images: A Beginner's Guide to Set Theory

[Romono]

Could someone please explain how the image of a set A' ⊆ A is the set: f(A') = {b | b = f(a) for some a ∈ A'}. And how can the complement of A be a subset of A? Forgive my ignorance here, I'm a beginning student of set theory.

Edit: Maybe I should rephrase my question: Could you explain what "the image of a set A' ⊆ A is the set: f(A') = {b | b = f(a) for some a ∈ A'}" actually means? Could you break it down? I don't understand what an image of a set is even after reading the definition here.

 

[Blazejr]

Regarding complement of a set:
You can only take complement of a set B relative to its superset (A is superset of B if B is a subset of A). Complement of B relative to A is set of all elements of A that are not in B.
For example: Let A be set of all natural numbers: 1,2,3,... and B be set of all even numbers: 2,4,6,...
Then complement of B relative to A would be all odd numbers: 1,3,5,...
However we could also regard B (set of even natural numbers) as subset of set of all reals. Then it's complement (relative to set of real numbers) would be all real numbers except even naturals.

It seems absurd at first glance that complement of a set A can be its subset but it is indeed possible. Complement of A relative to itself is set of all elements of A that are not in A. Obviously there are no such elements, so it is empty set. Empty set is subset of every set by definition of being a subset (C is subset of D if and only if all elements of C belong to D).

About the image of a set: I assume that you already know what a function is. Let f be function from set A to set B. We call A the domain of f and B the codomain. Note that function might not be surjective, that is there might be elements y in B such that there is no x in A such that f(x)=y.
For example: Let A be set of all natural numbers and B be set of all natural numbers and f: A->B be defined by f(x)=2x. Then clearly for every odd y there is no x such that f(x)=y.
We call the set of all y in B such that there is x in A such that f(x)=y the range of A. So in example above range of f would be all even numbers.

f is a function defined on set A. However we can ask what is the range of f if we act with it on just some subset of A? For example, let A' be set of all numbers divisible by 3 and f be defined as above. Then range of A restricted to A' would be set of all numbers divisible by 6. We call that set image of A' in f.