$V$ is a vector space, $W,W',U,U'$ are subspaces of $V$ prove or disprove the following: A) if $U+W=U'+W'$ then $W=W' , U=U'$ or $W=U' , W'=U$  
B) if $U+W=V$ then for all $Y\subset V$ :  $Y=(Y\cap U)+(Y\cap W)$
I need help of how to approach these types of questions, I have tried to bring a counter example for A but I didn't get it, but I still also don't know how to prove it or if I just failed bringing the example.
 any explanation, tips, hints are appreciated, thanks in advance.
 A: Your intuition was correct about part A: there is a counterexample! Learning to construct counterexamples is a skill that takes practice. I'll try to explain my thought process for questions like these; I hope it helps.
This statement in particular might seem daunting because the consequent of the implication is a disjunction of two conjunctions, so it might seem hard to understand when this consequent will be false (which is what we need to understand -- a counterexample to an implication is just a situation where the antecedent is true and the consequent is false).
So, the first step should be to try to understand what the implication is "really saying". By this I mean you need to digest and internalize the content of this implication, which often involves rephrasing the statement in a more intuitive way. For me, this implication is "really saying" that "it's impossible to write a subspace as a sum of two subspaces in two different ways". This sentence is not meant to be a precise mathematical statement, but it is meant to be more intuitive and easy to think about. Does it make sense that this is just an imprecise rephrasing of the implication? If not, can you come up with a rephrasing that's easier for you to think about?
Once you've internalized what the statement is trying to say, we can try to produce a counterexample. A counterexample to this claim, based on my intuitive understanding, would be produced by writing a subspace of a vector space as a sum of two subspaces is two distinct ways. Can you think of an example of this happening? Once you have such an example, you can go back and check that it really is a counterexample to the precise statement you were given.
As you get more experience, this process of internalizing the content of mathematical statements and producing counterexamples will become easier and easier.
