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D. Prove or disprove: Let A, B, and C be sets. If A = B - C, then B = = AUC.

D. Prove or disprove: Let A, B, and C be sets. If A = B - C, then B = = AUC.

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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Please solve the following discrete problem 

**Problem D: Prove or Disprove** Let \( A \), \( B \), and \( C \) be sets. If \( A = B - C \), then \( B = A \cup C \). ### Explanation: The problem asks you to either prove that the given statement is true for all sets \( A \), \( B \), and \( C \), or provide a counterexample to show that it is false. - **\( B - C \):** This represents the set of elements that are in \( B \) but not in \( C \). - **\( A = B - C \):** This means that \( A \) contains exactly the elements that are in \( B \) but not in \( C \). - **\( B = A \cup C \):** This proposes that \( B \) should be made up of all elements in \( A \) along with all elements in \( C \). Consider what it means for \( A \) to be equal to \( B - C \) and whether this logically leads to \( B \) being equal to \( A \cup C \). You can use examples with specific sets to test whether this holds true in all cases.
Transcribed Image Text:**Problem D: Prove or Disprove** Let \( A \), \( B \), and \( C \) be sets. If \( A = B - C \), then \( B = A \cup C \). ### Explanation: The problem asks you to either prove that the given statement is true for all sets \( A \), \( B \), and \( C \), or provide a counterexample to show that it is false. - **\( B - C \):** This represents the set of elements that are in \( B \) but not in \( C \). - **\( A = B - C \):** This means that \( A \) contains exactly the elements that are in \( B \) but not in \( C \). - **\( B = A \cup C \):** This proposes that \( B \) should be made up of all elements in \( A \) along with all elements in \( C \). Consider what it means for \( A \) to be equal to \( B - C \) and whether this logically leads to \( B \) being equal to \( A \cup C \). You can use examples with specific sets to test whether this holds true in all cases.
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