I have several preimage and image statements to compare and I've completed the comparison part but I also need to give an example where the inclusion is proper.
1) $ $f [$f^{-1}$[B]]$ $ $ \subseteq B$
2) $B \subseteq $ $f^{-1}[f$[B]]
3) f [$\bigcap$ $A_t$] $ \subseteq $ $\bigcap$[f [$A_t$]
4) f [$A_1$] \ f [$A_2$ $ \subseteq $ f [$A_1$ \ $A_2$]
[$A_t$] is an indexed family.
I do know that to be a proper class one class is strictly contained within a larger class and excludes some of its members. Any advice would be greatly appreciated!
Are you sure about number 1?
In another of your questions, Injective Equivalence, you have already shown that for conditions (2) and (3), equality holds just in the case where $f$ is injective. So, for examples where the inclusion is strict, look for functions which are not injective.
(The simplest non-injective function is the function $f:\{0,1\}\to \{a\}$ given by $f(0)=f(1)=a$. This is useful to know when constructing counterexamples.)
Number 4 is similar.
As an aside, your terminology here is a bit off:
You mean a proper subset (or occasionally a proper subclass). A proper class is something altogether different.
Another way to define a proper subset is to say that $A$ is a proper subset of $B$ if $A\subseteq B$ and $A\neq B$. For this reason, some people use the notation $A\subsetneq B$ (to avoid the ambiguity over "$\subset$" - see here: $\subset$ vs $\subseteq$ when *not* referring to strict inclusion).