The image asks us to prove the set identity:

$A - B = A \cap B^c = B - A^c$

Let's break it down step by step.

1. First Equality: $A - B = A \cap B^c$

• Definition of $A - B$: The difference of two sets, $A - B$, consists of all elements that are in $A$ but not in $B$. So, $A - B = \{x \mid x \in A \text{ and } x \notin B\}.$

• Definition of $A \cap B^c$: The intersection of $A$ and the complement of $B$, $B^c$, consists of all elements that are in both $A$ and $B^c$. By definition, $B^c$ contains all elements not in $B$, so: $A \cap B^c = \{x \mid x \in A \text{ and } x \notin B\}.$

Thus, $A - B = A \cap B^c$, because both represent the set of elements that are in $A$ but not in $B$.

2. Second Equality: $A \cap B^c = B - A^c$

• Definition of $B - A^c$: The set $B - A^c$ consists of all elements that are in $B$ but not in $A^c$ (the complement of $A$). By definition, $A^c$ contains all elements not in $A$, so: $B - A^c = \{x \mid x \in B \text{ and } x \notin A^c\}.$ Since $x \notin A^c$ means $x \in A$, we have: $B - A^c = \{x \mid x \in B \text{ and } x \in A\} = A \cap B.$

Thus, $A \cap B^c = B - A^c$, completing the proof.

Conclusion:

$A - B = A \cap B^c = B - A^c$

Let me know if you would like more details or clarifications!

Here are five related questions for further exploration:

1. What is the complement of a set, and how does it relate to set differences?
2. How would you prove the set identity $A - (B \cap C) = (A - B) \cup (A - C)$?
3. How do De Morgan’s laws help in understanding set complements and unions/intersections?
4. Can you provide an example using specific sets to verify this identity numerically?
5. What happens if sets $A$ and $B$ are disjoint? Does the identity still hold?

Tip: Understanding Venn diagrams can significantly help in visualizing set operations like union, intersection, and complement!