What is the trick to identify which of these are true I was hoping to not have to create a bunch of fictional sets so that I can solve this problem. Any trick or rules to this?

The sets A and B are subsets of a universal set U. Which of the following relations is always true?
(a) $A'∩ B' ⊂ (A ∩ B)'$
(b) $B ∩ A ⊂ B$
(c) $(A ∪ B) ⊂ A' ∩ B'$
(d) $A ∪ B ⊂ A ∩ B$

 A: by De Morgan's laws: $A' ∩ B'=(A ∪ B)'$. I use this in (a) and (c)  
(a) True: $A' ∩ B'=(A ∪ B)'⊂ (A ∩ B)'$. when a set decreases its complement increases.
(b) True: $x\in B ∩ A $ means $x$ is in both $A$ and $B$. so $x\in B$.
(c) always false:  $(A ∪ B) ⊂ A' ∩ B'\iff (A ∪ B) ⊂(A ∪ B)'$. a set is disjoint from its complement.
(d) this is not necessarily true; for example $A=\{1,2\} $ and $B=\{2,3\}$. (in fact this is true iff $A=B$)
A: Draw the diagrams. And see which makes sense. 
A: Use Venn diagram for d (counterexample).
Prove b by definition of subset relation (take an arbitrary element in the left set, show that it belongs to the right.
Use de Morgan law for a and c. Then use above mentioned method.
A: If you're simply interested in quickly evaluating the truth or falsity of these sentences, rather than spelling out geometric or formal proofs, consider simply translating the statements into English. This can improve your intuition immensely.


*

*Statement (a) claims neither A nor B implies not both A and B (which is equivalent to either not A or not B).

*Statement (b) claims both A and B implies B.

*Statement (c) claims either A or B implies neither A nor B.

*Statement (d) claims either A or B implies both A and B.


These are informal translations, of course, and they can be made much more precise. For example, what statement (a) is really saying is that anything that is in neither A nor B is not in both A and B. Or, even more pedantically: the set of all things that are in neither A nor B is a subset of the set of all things that are not in both A and B. 
Nevertheless, being able to translate fluently in this way is important.
