The exercise you've provided is in set theory and function notation. Here's a breakdown of the tasks in the exercise:

We are given two sets $E$ and $F$, and a function $f : E \to F$. The goal is to prove certain properties involving subsets and preimages under the function $f$.

1. $A \subseteq E$, show that $A \subseteq f^{-1}(f(A))$

We need to prove that any set $A$ in the domain of $f$ is a subset of the preimage of the image of $A$.

2. $B \subseteq F$, show that $f(f^{-1}(B)) \subseteq B$

We need to show that for any set $B$ in the codomain, the image of the preimage of $B$ is a subset of $B$.

3. $A \subseteq E$, show that $f^{-1}(f(A)) = A$ if and only if $f$ is injective

This is asking to prove that the preimage of the image of $A$ is equal to $A$ if and only if $f$ is injective (one-to-one).

4. Show that $f$ is injective if and only if $\forall A, B \subseteq E$, $f(A \cap B) = f(A) \cap f(B)$

We need to prove that $f$ is injective if and only if it preserves intersections between sets.

5. Show that $f$ is injective if and only if $\forall A, B \subseteq E$, $f(A) \cap f(B) = \emptyset$ implies $A \cap f^{-1}(B) = \emptyset$

This is asking to show that $f$ being injective is equivalent to the property that disjoint images imply disjoint preimages.

Would you like to go through any specific part of these proofs in detail? Let me know!

Here are 5 related questions to expand on these concepts:

1. How is the image of a set defined under a function?
2. What is the definition of the preimage of a set?
3. What is the difference between injective, surjective, and bijective functions?
4. How does the preimage operation behave under intersections and unions?
5. Can a function be injective but not surjective? Explain.

Tip: When working with preimages and images, always clearly define the domain and codomain of your function to avoid confusion about where sets live.