Let f:A →B and let C⊆A
Prove that f(A) \ f(C) is a subset of f(A\C)

Question

anonymous · Answer

In case the characters in the question didnt show up properly:

Let 
$$f: A \rightarrow B$$ and let $$C \subseteq A$$
Prove that f(A) \ f(C) is a subset of f(A\C)

I never learned how to do proofs properly like I should've, and I'm not sure where to even start with this.

anonymous · Answer

To prove a set is a subset of another, you have to show that an element belonging to the first set implies that the element belongs to the second set.

For our benefit, does $f(A)\backslash f(C)$ denote all $f(x)\in f(A), f(x)\not\in f(C)$?

anonymous · Answer

I still don't know how to get the math editor to always do what I want, so sorry for any sloppiness. But yeah, I would assume that's what it means. 

f(A) \ f(C) = the set of all x such that f(x) in f(A), f(x) not in f(C), x in A. You didn't state the x in A, so maybe that is just given or doesn't need to be put? 

And f(A\C) Im not quite sure how to write. I know what it means, but how to type it out in terms of sets or anything, Im not sure. Maybe that's the problem. 

Since C is a subset of A, can I automatically assume f(C) is a subset of f(A)? This is just where I go in circles, lol.

anonymous · Answer

I'd say f(A\C) is the set
$$\left\{f(x)~:~x\in A,x\not\in C\right\}$$
(i.e. all function values of $x$ with $x$ in $A$ and not in $C$).

anonymous · Answer

So an element in the first set implies that the element must belong to the 2nd set. So an element in the first set would be:
$$\left\{ f(x): f(x) \in f(A), f(x) \notin f(C) \right\}$$
But what implication can I make from that? If f(x) is in f(A)....does that mean x has to be in A? Is that something I have to show, or is that implied, or is that even true at all?

anonymous · Answer

Start off by lettng $y=\in f(A)\backslash f(C)$ then $y\in f(A)$ and $y\not\in f(C)$.

Since $y\in f(A)$ by definition you know $\exists~a\in A$ with $y=f(a)$.

Since $y\not\in f(C)$ you know that $a\not\in C$. This means that $a\in A\backslash C$.

Thus $y=f(A)\in f(A\backslash C)$ by the definition of $f(A\backslash C)$ (which I posted earlier $\left\{f(x)~:~x\in A,x\not\in C\right\}$.

anonymous · Answer

Ah, okay. I think that was just a step I wasn't sure if I was allowed to make, but you said by definition, so guess so. Alright, thanks ^_^