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Technical Question: Which of the following best characterizes the statements, where A and B are any set Α) ΦΕ Α А B) A\BnB = Ø A. Only A is TRUE B. Only B is TRUE C. Both A and B are TRUE 1 D. Both A and B are FALSE

Technical Question: Which of the following best characterizes the statements, where A and B are any set Α) ΦΕ Α А B) A\BnB = Ø A. Only A is TRUE B. Only B is TRUE C. Both A and B are TRUE 1 D. Both A and B are FALSE

Advanced Engineering Mathematics
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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### Technical Question: Set Theory Analysis --- **Question:** Which of the following best characterizes the statements, where A and B are any set? ### Statements: **A)** \( \varnothing \in A \) **B)** \( A \setminus B \cap B = \varnothing \) --- ### Options: A. Only A is TRUE B. Only B is TRUE C. Both A and B are TRUE D. Both A and B are FALSE ### Explanation of Statements: 1. **Statement A: \( \varnothing \in A \)** This statement claims that the empty set is an element of set A. It is important to distinguish between the empty set being an element of a set and being a subset of a set. The notation \( \varnothing \in A \) means that the empty set itself is one of the elements in set A, not that it’s a subset of A. 2. **Statement B:** \( A \setminus B \cap B = \varnothing \) This statement describes the intersection between elements in \( A \) excluding those in \( B \) and the set \( B \). Essentially, \( A \setminus B \) takes all elements in \( A \) that are not in \( B \), and the intersection with \( B \) should logically be empty, because any element that is in \( A \setminus B \) cannot be in \( B \). ### Visualization: To better understand, consider two different sets, - A: The set comprising elements of set {1, 2, 3} - B: The set comprising elements of set {2, 3} When considering statement **B**, we take \( A \setminus B \) which results in set {1}, then intersecting this result with \( B \) {2,3} will give an empty set. Thus, statement B holds true in general if A and B are any sets. ### Evaluation of Options: - **Option A (Only A is TRUE):** This option would mean the empty set is an element of A. While feasible in some contexts, it does not hold universally for any set A. - **Option B (Only B is TRUE):** This indicates that the intersection of \( A \setminus B \) and \( B \) is the empty set, which is always true for any sets A and B for the reasons
Transcribed Image Text:### Technical Question: Set Theory Analysis --- **Question:** Which of the following best characterizes the statements, where A and B are any set? ### Statements: **A)** \( \varnothing \in A \) **B)** \( A \setminus B \cap B = \varnothing \) --- ### Options: A. Only A is TRUE B. Only B is TRUE C. Both A and B are TRUE D. Both A and B are FALSE ### Explanation of Statements: 1. **Statement A: \( \varnothing \in A \)** This statement claims that the empty set is an element of set A. It is important to distinguish between the empty set being an element of a set and being a subset of a set. The notation \( \varnothing \in A \) means that the empty set itself is one of the elements in set A, not that it’s a subset of A. 2. **Statement B:** \( A \setminus B \cap B = \varnothing \) This statement describes the intersection between elements in \( A \) excluding those in \( B \) and the set \( B \). Essentially, \( A \setminus B \) takes all elements in \( A \) that are not in \( B \), and the intersection with \( B \) should logically be empty, because any element that is in \( A \setminus B \) cannot be in \( B \). ### Visualization: To better understand, consider two different sets, - A: The set comprising elements of set {1, 2, 3} - B: The set comprising elements of set {2, 3} When considering statement **B**, we take \( A \setminus B \) which results in set {1}, then intersecting this result with \( B \) {2,3} will give an empty set. Thus, statement B holds true in general if A and B are any sets. ### Evaluation of Options: - **Option A (Only A is TRUE):** This option would mean the empty set is an element of A. While feasible in some contexts, it does not hold universally for any set A. - **Option B (Only B is TRUE):** This indicates that the intersection of \( A \setminus B \) and \( B \) is the empty set, which is always true for any sets A and B for the reasons
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