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B are ses, -) A-Β=ΑnΒ . ) (ΑnΒ )υ (ΑnΒ ) = Α.

B are ses, -) A-Β=ΑnΒ . ) (ΑnΒ )υ (ΑnΒ ) = Α.

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 Exploration** In this section, we will address two fundamental properties involving set operations. Consider \( A \) and \( B \) as sets. We are tasked with demonstrating the following: a) \( A - B = A \cap \overline{B} \) This equation signifies that the difference between sets \( A \) and \( B \) is equivalent to the intersection of set \( A \) and the complement of set \( B \). In set terms, it implies that elements in \( A \) that are not in \( B \) are precisely the elements that are common to \( A \) and not in \( B \). b) \( (A \cap B) \cup (A \cap \overline{B}) = A \) This equation indicates that the union of the intersection of \( A \) and \( B \) with the intersection of \( A \) and the complement of \( B \) equals set \( A \). Essentially, this means all elements belonging to \( A \) will be accounted for by including elements common to \( A \) and \( B \), and those common to \( A \) but not in \( B \). These identities are crucial for understanding the relationships and operations involving sets and their complements.
Transcribed Image Text:**Set Theory Exploration** In this section, we will address two fundamental properties involving set operations. Consider \( A \) and \( B \) as sets. We are tasked with demonstrating the following: a) \( A - B = A \cap \overline{B} \) This equation signifies that the difference between sets \( A \) and \( B \) is equivalent to the intersection of set \( A \) and the complement of set \( B \). In set terms, it implies that elements in \( A \) that are not in \( B \) are precisely the elements that are common to \( A \) and not in \( B \). b) \( (A \cap B) \cup (A \cap \overline{B}) = A \) This equation indicates that the union of the intersection of \( A \) and \( B \) with the intersection of \( A \) and the complement of \( B \) equals set \( A \). Essentially, this means all elements belonging to \( A \) will be accounted for by including elements common to \( A \) and \( B \), and those common to \( A \) but not in \( B \). These identities are crucial for understanding the relationships and operations involving sets and their complements.
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