The image contains a set theory problem in the Indonesian language. Here's a translation of the problem and an outline of how we can prove each part.

Given: $A$, $B$, and $C$ are sets. Prove the following:

(a) $(A - C) \cap (C - B) = \emptyset$

Explanation:

• $A - C$ represents the set of elements in $A$ but not in $C$.
• $C - B$ represents the set of elements in $C$ but not in $B$.
• The intersection of $(A - C)$ and $(C - B)$ is empty, meaning there are no elements that belong to both $(A - C)$ and $(C - B)$.

(b) $(B - A) \cup (C - A) = (B \cup C) - A$

Explanation:

• $B - A$ is the set of elements in $B$ but not in $A$.
• $C - A$ is the set of elements in $C$ but not in $A$.
• The union of $(B - A)$ and $(C - A)$ should be equal to $(B \cup C) - A$, which is the set of elements in $B \cup C$ but not in $A$.

(c) $(A - B) - C \subseteq A - C$

Explanation:

• $(A - B)$ is the set of elements in $A$ but not in $B$.
• $(A - B) - C$ is the set of elements in $(A - B)$ but not in $C$.
• The goal is to prove that $(A - B) - C$ is a subset of $A - C$.

Let's go through each part step by step and prove these statements.

(a) Proof of $(A - C) \cap (C - B) = \emptyset$:

• If $x \in (A - C)$, then $x \in A$ and $x \notin C$.
• If $x \in (C - B)$, then $x \in C$ and $x \notin B$.
• For $x$ to belong to both sets, $x$ would need to be in $A$ and not in $C$, but also in $C$ and not in $B$, which is a contradiction because $x$ cannot be both in and not in $C$ at the same time.
• Hence, $(A - C) \cap (C - B) = \emptyset$.

(b) Proof of $(B - A) \cup (C - A) = (B \cup C) - A$:

• $(B - A) \cup (C - A)$ contains elements that are either in $B$ but not in $A$ or in $C$ but not in $A$.
• $(B \cup C) - A$ contains elements that are in $B \cup C$ but not in $A$, meaning they are either in $B$ or $C$, but not in $A$.
• These two descriptions are equivalent, so $(B - A) \cup (C - A) = (B \cup C) - A$.

(c) Proof of $(A - B) - C \subseteq A - C$:

• If $x \in (A - B) - C$, then $x \in (A - B)$ and $x \notin C$.
• This means $x \in A$, $x \notin B$, and $x \notin C$.
• If $x \in A$ and $x \notin C$, then $x \in A - C$.
• Therefore, every element of $(A - B) - C$ is also in $A - C$, meaning $(A - B) - C \subseteq A - C$.

Would you like further details on any of these proofs?

Here are five related questions you might explore:

1. How would the proofs change if the sets $A$, $B$, or $C$ were empty?
2. Can you think of a real-world example that models these set operations?
3. What happens if we replace $A - B$ with $B - A$ in part (c)?
4. How does De Morgan's Law relate to these set identities?
5. Can Venn diagrams help visualize these set operations for better understanding?

Tip: To gain intuition about set operations, try drawing Venn diagrams for the sets involved. This helps in visualizing how the sets interact.