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4. Suppose there are three sets A, B,C. (a) List all the possibilities for where an element could be. For exam- ple, an element could belong to A and B but not C. (Label the possibilities 1 through however many there are) (b) Draw a generic Venn diagram for three sets, and show where each of your labeled possibilities will be located.

4. Suppose there are three sets A, B,C. (a) List all the possibilities for where an element could be. For exam- ple, an element could belong to A and B but not C. (Label the possibilities 1 through however many there are) (b) Draw a generic Venn diagram for three sets, and show where each of your labeled possibilities will be located.

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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**Problem 4: Venn Diagram Analysis for Three Sets** Suppose there are three sets \(A\), \(B\), and \(C\). **(a) Listing Possibilities:** An element can belong to various combinations of these sets. The possible scenarios include: 1. \(A \cap B \cap C\) (The element is in all three sets) 2. \(A \cap B \cap \overline{C}\) (The element is in \(A\) and \(B\) but not in \(C\)) 3. \(A \cap \overline{B} \cap C\) (The element is in \(A\) and \(C\) but not in \(B\)) 4. \(\overline{A} \cap B \cap C\) (The element is in \(B\) and \(C\) but not in \(A\)) 5. \(A \cap \overline{B} \cap \overline{C}\) (The element is only in \(A\)) 6. \(\overline{A} \cap B \cap \overline{C}\) (The element is only in \(B\)) 7. \(\overline{A} \cap \overline{B} \cap C\) (The element is only in \(C\)) 8. \(\overline{A} \cap \overline{B} \cap \overline{C}\) (The element is in none of the sets) Label these possibilities as 1 through 8. **(b) Venn Diagram Description:** A generic Venn diagram for three sets consists of three overlapping circles. Each circle represents one of the sets \(A\), \(B\), or \(C\), and the overlapping regions illustrate the intersections between the sets: - The central region where all three circles overlap is labeled as possibility 1, \(A \cap B \cap C\). - The regions where only two circles overlap are labeled: - The overlap between \(A\) and \(B\) only (excluding \(C\)) is labeled as possibility 2. - The overlap between \(A\) and \(C\) only (excluding \(B\)) is labeled as possibility 3. - The overlap between \(B\) and \(C\) only (excluding \(A\))
Transcribed Image Text:**Problem 4: Venn Diagram Analysis for Three Sets** Suppose there are three sets \(A\), \(B\), and \(C\). **(a) Listing Possibilities:** An element can belong to various combinations of these sets. The possible scenarios include: 1. \(A \cap B \cap C\) (The element is in all three sets) 2. \(A \cap B \cap \overline{C}\) (The element is in \(A\) and \(B\) but not in \(C\)) 3. \(A \cap \overline{B} \cap C\) (The element is in \(A\) and \(C\) but not in \(B\)) 4. \(\overline{A} \cap B \cap C\) (The element is in \(B\) and \(C\) but not in \(A\)) 5. \(A \cap \overline{B} \cap \overline{C}\) (The element is only in \(A\)) 6. \(\overline{A} \cap B \cap \overline{C}\) (The element is only in \(B\)) 7. \(\overline{A} \cap \overline{B} \cap C\) (The element is only in \(C\)) 8. \(\overline{A} \cap \overline{B} \cap \overline{C}\) (The element is in none of the sets) Label these possibilities as 1 through 8. **(b) Venn Diagram Description:** A generic Venn diagram for three sets consists of three overlapping circles. Each circle represents one of the sets \(A\), \(B\), or \(C\), and the overlapping regions illustrate the intersections between the sets: - The central region where all three circles overlap is labeled as possibility 1, \(A \cap B \cap C\). - The regions where only two circles overlap are labeled: - The overlap between \(A\) and \(B\) only (excluding \(C\)) is labeled as possibility 2. - The overlap between \(A\) and \(C\) only (excluding \(B\)) is labeled as possibility 3. - The overlap between \(B\) and \(C\) only (excluding \(A\))
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