Math Problem Statement

Prove that A - B = A ∩ B^c = B - A^c

Solution

The image asks us to prove the set identity:

AB=ABc=BAcA - B = A \cap B^c = B - A^c

Let's break it down step by step.

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

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

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

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

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

  • Definition of BAcB - A^c: The set BAcB - A^c consists of all elements that are in BB but not in AcA^c (the complement of AA). By definition, AcA^c contains all elements not in AA, so: BAc={xxB and xAc}.B - A^c = \{x \mid x \in B \text{ and } x \notin A^c\}. Since xAcx \notin A^c means xAx \in A, we have: BAc={xxB and xA}=AB.B - A^c = \{x \mid x \in B \text{ and } x \in A\} = A \cap B.

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

Conclusion:

AB=ABc=BAcA - 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(BC)=(AB)(AC)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 AA and BB are disjoint? Does the identity still hold?

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

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Math Problem Analysis

Mathematical Concepts

Set Theory
Set Operations
Complement of a Set
Set Difference

Formulas

A - B = {x | x ∈ A and x ∉ B}
A ∩ B^c = {x | x ∈ A and x ∉ B}
B - A^c = {x | x ∈ B and x ∉ A^c}

Theorems

Basic Set Identities
Set Difference and Complement

Suitable Grade Level

Grades 10-12 (Advanced High School / Undergraduate level)