11. For sets A and B, define AAB=(A\B) U (B\A). (a) Let A and B be sets. Prove that A = B if and only if AAB = 0. (b) Let A and B be sets. Prove that AAB = BAA. (c) Let A, B, and C be sets. Prove that (AAB) AC = AA (BAC).
11. For sets A and B, define AAB=(A\B) U (B\A). (a) Let A and B be sets. Prove that A = B if and only if AAB = 0. (b) Let A and B be sets. Prove that AAB = BAA. (c) Let A, B, and C be sets. Prove that (AAB) AC = AA (BAC).
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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![**11. Set Operations and Properties**
For sets \( A \) and \( B \), the symmetric difference is defined as:
\[ A \triangle B = (A \setminus B) \cup (B \setminus A). \]
**Exercises:**
(a) Let \( A \) and \( B \) be sets. Prove that \( A = B \) if and only if \( A \triangle B = \emptyset \).
(b) Let \( A \) and \( B \) be sets. Prove that \( A \triangle B = B \triangle A \).
(c) Let \( A \), \( B \), and \( C \) be sets. Prove that \( (A \triangle B) \triangle C = A \triangle (B \triangle C) \).
**Explanation of Concepts:**
- \( A \setminus B \) denotes the set of elements that are in \( A \) but not in \( B \).
- \( B \setminus A \) denotes the set of elements that are in \( B \) but not in \( A \).
- The symmetric difference \( A \triangle B \) consists of elements that are in either of the sets \( A \) or \( B \), but not in their intersection.
- The symbol \( \cup \) represents the union of two sets.
- \( \emptyset \) denotes the empty set, which has no elements.
**Properties to Prove:**
1. **Identity**: Two sets \( A \) and \( B \) are identical if their symmetric difference is empty.
2. **Commutativity**: The symmetric difference operation is commutative.
3. **Associativity**: The symmetric difference operation is associative, allowing for regrouping without affecting the result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7626d59-6405-4948-a625-19aca32c9eb7%2Ff25b55f8-02c0-4c27-b5d2-111f71896187%2Fxb4nbb9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**11. Set Operations and Properties**
For sets \( A \) and \( B \), the symmetric difference is defined as:
\[ A \triangle B = (A \setminus B) \cup (B \setminus A). \]
**Exercises:**
(a) Let \( A \) and \( B \) be sets. Prove that \( A = B \) if and only if \( A \triangle B = \emptyset \).
(b) Let \( A \) and \( B \) be sets. Prove that \( A \triangle B = B \triangle A \).
(c) Let \( A \), \( B \), and \( C \) be sets. Prove that \( (A \triangle B) \triangle C = A \triangle (B \triangle C) \).
**Explanation of Concepts:**
- \( A \setminus B \) denotes the set of elements that are in \( A \) but not in \( B \).
- \( B \setminus A \) denotes the set of elements that are in \( B \) but not in \( A \).
- The symmetric difference \( A \triangle B \) consists of elements that are in either of the sets \( A \) or \( B \), but not in their intersection.
- The symbol \( \cup \) represents the union of two sets.
- \( \emptyset \) denotes the empty set, which has no elements.
**Properties to Prove:**
1. **Identity**: Two sets \( A \) and \( B \) are identical if their symmetric difference is empty.
2. **Commutativity**: The symmetric difference operation is commutative.
3. **Associativity**: The symmetric difference operation is associative, allowing for regrouping without affecting the result.
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