https://file.aiquickdraw.com/mathbot/file/1725540545296qdpik0wg.jpg

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!

Learn how to prove the set identity A - B = A ∩ B^c = B - A^c, involving key set theory concepts such as set difference, intersection, and complement. Suitable for advanced high school and undergraduate 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