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4. If A and B are ANY two sets with AB, determine the truth-values of the following statements. If a statement is false, give specific examples of sets A and B that serve as a counter-example a. If (B\A) = Ø, then ACB b. If ACB, then (B\A) #Ø

4. If A and B are ANY two sets with AB, determine the truth-values of the following statements. If a statement is false, give specific examples of sets A and B that serve as a counter-example a. If (B\A) = Ø, then ACB b. If ACB, then (B\A) #Ø

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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### Set Theory Problem #### Problem 4: If \( A \) and \( B \) are ANY two sets with \( A \subseteq B \), determine the truth-values of the following statements. If a statement is false, give specific examples of sets \( A \) and \( B \) that serve as a counter-example. a. If \( (B \setminus A) = \emptyset \), then \( A \subseteq B \) b. If \( A \subseteq B \), then \( (B \setminus A) \neq \emptyset \) #### Explanation: - For statement (a), the notation \( (B \setminus A) \) denotes the set difference of \( B \) and \( A \), which consists of elements in \( B \) that are not in \( A \). \( \emptyset \) represents the empty set. The statement checks if the difference being empty implies \( A \subseteq B \). - For statement (b), the notation \( A \subseteq B \) indicates that \( A \) is a subset of \( B \). The statement checks if this subset relation means the set difference \( (B \setminus A) \) is non-empty. Evaluate each statement to verify its truth or provide counter-examples where applicable.
Transcribed Image Text:### Set Theory Problem #### Problem 4: If \( A \) and \( B \) are ANY two sets with \( A \subseteq B \), determine the truth-values of the following statements. If a statement is false, give specific examples of sets \( A \) and \( B \) that serve as a counter-example. a. If \( (B \setminus A) = \emptyset \), then \( A \subseteq B \) b. If \( A \subseteq B \), then \( (B \setminus A) \neq \emptyset \) #### Explanation: - For statement (a), the notation \( (B \setminus A) \) denotes the set difference of \( B \) and \( A \), which consists of elements in \( B \) that are not in \( A \). \( \emptyset \) represents the empty set. The statement checks if the difference being empty implies \( A \subseteq B \). - For statement (b), the notation \( A \subseteq B \) indicates that \( A \) is a subset of \( B \). The statement checks if this subset relation means the set difference \( (B \setminus A) \) is non-empty. Evaluate each statement to verify its truth or provide counter-examples where applicable.
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