https://file.aiquickdraw.com/mathbot/file/1728498079687zvu4o15r.jpg

The statement to prove is: 

\[
(A - B) \cup (A \cap B) = A
\]

### Proof:

1. **Left-hand side:** The expression \( (A - B) \cup (A \cap B) \) represents the union of two sets:
   - \( A - B \) is the set of elements in \( A \) but not in \( B \).
   - \( A \cap B \) is the set of elements common to both \( A \) and \( B \).

2. **Right-hand side:** The expression on the right-hand side is simply \( A \), the set of all elements in \( A \).

3. **Objective:** Show that \( (A - B) \cup (A \cap B) = A \).

### Step-by-step breakdown:

#### (i) Proving \( (A - B) \cup (A \cap B) \subseteq A \):
- If an element \( x \in (A - B) \), then by definition, \( x \in A \) and \( x \notin B \). Therefore, \( x \in A \).
- If an element \( x \in (A \cap B) \), then by definition, \( x \in A \) and \( x \in B \). Therefore, \( x \in A \).
- In both cases, whether \( x \) belongs to \( A - B \) or \( A \cap B \), we conclude that \( x \in A \).
- Thus, any element in \( (A - B) \cup (A \cap B) \) must be in \( A \). Therefore, \( (A - B) \cup (A \cap B) \subseteq A \).

#### (ii) Proving \( A \subseteq (A - B) \cup (A \cap B) \):
- Let \( x \in A \). There are two possibilities:
   - Either \( x \in B \) or \( x \notin B \).
   - If \( x \in B \), then \( x \in A \cap B \), so \( x \in (A \cap B) \subseteq (A - B) \cup (A \cap B) \).
   - If \( x \notin B \), then \( x \in A - B \), so \( x \in (A - B) \subseteq (A - B) \cup (A \cap B) \).
- In both cases, \( x \in (A - B) \cup (A \cap B) \).
- Therefore, \( A \subseteq (A - B) \cup (A \cap B) \).

### Conclusion:
Since we have shown that \( (A - B) \cup (A \cap B) \subseteq A \) and \( A \subseteq (A - B) \cup (A \cap B) \), it follows that:
\[
(A - B) \cup (A \cap B) = A
\]

This completes the proof.

---

Would you like further details or have any questions on specific steps?

Here are five related questions to explore:
1. What is the significance of the set difference \( A - B \) in set theory?
2. How does the intersection \( A \cap B \) interact with the union operator?
3. Can this proof be generalized for more than two sets?
4. What is the meaning of the symmetric difference between sets?
5. How does the distributive property of sets relate to this proof?

**Tip:** In set theory, visualizing sets using Venn diagrams can help intuitively understand unions, intersections, and set differences.

Let A and B be two sets in a given universe. Prove the following: (A - B) ∪ (A ∩ B) = A.

Learn how to prove that (A - B) ∪ (A ∩ B) = A in set theory. This detailed solution explains key set theory concepts such as set difference, intersection, and union. Suitable for undergraduate mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation