Please help me with this.Use element argument to prove the statement. Assume that all sets are subsets of a universal set U, Statement: For all sets A, B, and C, A ∩ (B − C) = (A ∩ B) − (A ∩ C). Please do the proof like the way the example proof does it. Thanks. Certainly! Below is a transcription of the given image text, suitable for an educational website:---### StatementFor all sets \(A\), \(B\), and \(C\):\[(A - B) \cap (C - B) = (A \cap C) - B\]### ProofSuppose \(A\), \(B\), and \(C\) are any sets. To show that \((A - B) \cap (C - B) = (A \cap C) - B\), we must show that \((A - B) \cap (C - B) \subseteq (A \cap C) - B\) and that \((A \cap C) - B \subseteq (A - B) \cap (C - B)\).#### Part 1: Proof that \((A - B) \cap (C - B) \subseteq (A \cap C) - B\)Consider the sentences in the following scrambled list:1. By definition of set difference, \(x \in A\) and \(x \notin B\) and \(x \in C\) and \(x \notin B\).2. Thus \(x \in A\) and \(x \in C\) by definition of intersection and, in addition, \(x \notin B\).3. Therefore \(x \in (A \cap C) - B\) by the definition of set difference.4. By definition of intersection, \(x \in A\) and \(x \in B\) and \(x \in C\) and \(x \notin B\).To prove Part 1, select sentences from the list and put them in the correct order.1. Suppose \(x \in (A - B) \cap (C - B)\).2. By definition of intersection, \(x \in A - B\) and \(x \in C - B\).3. By definition of set difference, \(x \in A\) and \(x \notin B\) and \(x \in C\) and \(x \notin B\).4. Thus \(x \in A\) and \(x \in C\) by definition of intersection and, in addition, \(x \notin B\).5. Therefore \(x \in (A

Answered: Image /qna-images/answer/a7edf0f2-fa41-430c-83e9-3fbdf31d810a.jpg

Please help me with this. Use element argument to prove the statement. Assume that all sets are subsets of a universal set U, Statement: For all sets A, B, and C, A ∩ (B − C) = (A ∩ B) − (A ∩ C). Please do the proof like the way the example proof does it. Thanks.

Please help me with this. Use element argument to prove the statement. Assume that all sets are subsets of a universal set U, Statement: For all sets A, B, and C, A ∩ (B − C) = (A ∩ B) − (A ∩ C). Please do the proof like the way the example proof does it. Thanks.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please help me with this.

Use element argument to prove the statement. Assume that all sets are subsets of a universal set U, Statement: For all sets A, B, and C, A ∩ (B − C) = (A ∩ B) − (A ∩ C). Please do the proof like the way the example proof does it. Thanks.

Certainly! Below is a transcription of the given image text, suitable for an educational website:

---

### Statement

For all sets \(A\), \(B\), and \(C\):

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

### Proof

Suppose \(A\), \(B\), and \(C\) are any sets. To show that \((A - B) \cap (C - B) = (A \cap C) - B\), we must show that \((A - B) \cap (C - B) \subseteq (A \cap C) - B\) and that \((A \cap C) - B \subseteq (A - B) \cap (C - B)\).

#### Part 1: Proof that \((A - B) \cap (C - B) \subseteq (A \cap C) - B\)

Consider the sentences in the following scrambled list:

1. By definition of set difference, \(x \in A\) and \(x \notin B\) and \(x \in C\) and \(x \notin B\).
2. Thus \(x \in A\) and \(x \in C\) by definition of intersection and, in addition, \(x \notin B\).
3. Therefore \(x \in (A \cap C) - B\) by the definition of set difference.
4. By definition of intersection, \(x \in A\) and \(x \in B\) and \(x \in C\) and \(x \notin B\).

To prove Part 1, select sentences from the list and put them in the correct order.
1. Suppose \(x \in (A - B) \cap (C - B)\).
2. By definition of intersection, \(x \in A - B\) and \(x \in C - B\).
3. By definition of set difference, \(x \in A\) and \(x \notin B\) and \(x \in C\) and \(x \notin B\).
4. Thus \(x \in A\) and \(x \in C\) by definition of intersection and, in addition, \(x \notin B\).
5. Therefore \(x \in (A

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