[ (A \cup C) \cap (B \cup C) \subseteq (A \cap B) \cup C prove 

To prove the set-theoretic identity:

\[
(A \cup C) \cap (B \cup C) \subseteq (A \cap B) \cup C
\]

we will proceed by showing that if an element \( x \) belongs to the left-hand side, then it must also belong to the right-hand side.

### Proof:

1. **Assume \( x \in (A \cup C) \cap (B \cup C) \):**

   This means that:
   \[
   x \in (A \cup C) \quad \text{and} \quad x \in (B \cup C).
   \]

2. **Break down the implications of \( x \in (A \cup C) \):**
   
   \[
   x \in A \quad \text{or} \quad x \in C.
   \]
   
   Similarly, from \( x \in (B \cup C) \), we have:
   
   \[
   x \in B \quad \text{or} \quad x \in C.
   \]

3. **Consider two main cases:**

   - **Case 1: \( x \in C \)**

     If \( x \in C \), then clearly \( x \in (A \cap B) \cup C \) because \( x \in C \). Thus, \( x \) satisfies the right-hand side of the inclusion.

   - **Case 2: \( x \notin C \)**

     If \( x \notin C \), then from \( x \in (A \cup C) \), it must be that \( x \in A \). Similarly, from \( x \in (B \cup C) \), it must be that \( x \in B \). Therefore, \( x \in A \cap B \).

     Since \( x \in A \cap B \), it follows that \( x \in (A \cap B) \cup C \).

4. **Conclusion:**

   In both cases (whether \( x \in C \) or \( x \notin C \)), we have shown that \( x \in (A \cap B) \cup C \). Therefore, every element of \( (A \cup C) \cap (B \cup C) \) is also an element of \( (A \cap B) \cup C \), proving the inclusion:

   \[
   (A \cup C) \cap (B \cup C) \subseteq (A \cap B) \cup C.
   \]

---

Would you like further clarification or details on this proof?

Here are 5 related questions to extend your understanding:

1. What is the difference between union and intersection in set theory?
2. Can you think of a counterexample where the reverse inclusion does not hold?
3. How would the proof change if \( A \), \( B \), and \( C \) are disjoint sets?
4. What are the implications of this inclusion in terms of Venn diagrams?
5. How does the identity change if you replace union with symmetric difference?

**Tip:** When working with set identities, breaking down the sets using element-based proofs often makes things clearer.

(A ∪ C) ∩ (B ∪ C) ⊆ (A ∩ B) ∪ C prove

Explore the proof for the set-theoretic identity (A ∪ C) ∩ (B ∪ C) ⊆ (A ∩ B) ∪ C. This problem covers key concepts in set theory, including union, intersection, and subset relations, suitable for students in Grades 10-12.

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