Finding counterexamples in elementary set theory. I had the following two problems:

Find a counterexample for $f_*(A \cap B) \supseteq f_*(A) \cap f_*(B)$ and $ f_*(A-B) \subseteq f_*(A) -f_*(B).$ Where $f_*(X)$ is the image of $X$ under $f$ for some function $f:A \rightarrow B $ and some subset $X \subseteq A$.

I came with the following answer:
Let $ f: \mathbb{Z} \rightarrow \mathbb{Z} $, where $f: x\mapsto 0$; let $A=\{1,2\}$, and $B=\{3,4\}$.



*

*For $f_*(A \cap B) \supseteq f_*(A) \cap f_*(B)$, $$ f_*(A \cap B) = f_*\{ \varnothing\} = \varnothing \hspace{1cm} \text{and} \hspace{1cm} f_*(A) \cap f_*(B)=\{0\} \cap \{0\} = \{0\}. $$ It follows that $ \varnothing \nsupseteq \{0\} $.

*For $ f_*(A-B) \subseteq f_*(A) -f_*(B)$, $$f_*(A-B)=f_*(\{1,2\})=\{0\} \hspace{1cm} \text{and} \hspace{1cm} f_*(A) - f_*(B) = \{0\} - \{0\} = \varnothing$$ It follows that $ \{0\}  \nsubseteq \varnothing $.
Now (assuming my answer is correct), let's say I want to find more counterexamples or maybe a more general answer (e.g. all the conditions under which the inclusion fails). Other than by trial and error or just intuition, how would I do that? 
I know this is a broad question, so I'm specifically asking for methods or advice on how to build counterexamples in elementary set theory (for example, tips on where to begin or rules that apply in general that may be unobvious).
 A: Sets $A$ and $B$ and a function $f$ form a counterexample to $f_*(A\cap B)\supseteq f_*(A)\cap f_*(B)$ if and only if there is some $x\in f_*(A)\cap f_*(B)$ such that $x\notin f_*(A\cap B)$. This means that there must be some $a\in A$ and $b\in B$ such that $f(a)=f(b)=x$, but there is no $c\in A\cap B$ such that $f(c)=x$. In other words, we must have $a\ne b$. Thus, you can get every possible example by taking $A$ and $B$ to be non-empty sets, neither of which is a subset of the other, picking an $a\in A\setminus B$ and a $b\in B\setminus A$, and defining $f$ so that $f(a)=f(b)$, but $f(c)\ne f(a)$ whenever $c\in A\cap B$.
I’ll stop there for now in case you’d like to try to analyze the second problem in similar fashion on your own.
A: Your answers are correct and as you already stated, there is no general rule to produce counterexamples to the converse of a given result. However, it is often helpful to look at the proof of the correct result and study closely how the premises come into play. At every step where they're used, you may then try to produce an example - that doesn't satisfy the premise that is used - and see if this already suffices to produce a 'counterexample'.
Given that many set theoretical arguments are pretty forgiving in the sense that changing the sets just a bit won't change the outcome (this isn't true for the question that you were considering, but it is a common theme nonetheless), it is also advisable to first try to violate the premises of a given result heavily. Depending on the context, that could mean to violate it on a stationary set or a set of full measure or on a dense set or ... There is really no common rule, but in many situations there are 'natural candidates' to look at. Over time, you will develop an intuition for these things - at least in some situations.
In general, finding counterexamples can be incredibly difficult and often requires a lot of experience with a given subject.
