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Use the given information to fill in the number of elements for each region in the Venn diagram. n(A U B)=26, n(AB) = 2, n(A) = 14, and n(A' U B') = 47 The number of elements in region x is Z IS and the number of elements in region w is (Type whole numbers.) A W the number of elements in region y is m OU the number of elements in

Use the given information to fill in the number of elements for each region in the Venn diagram. n(A U B)=26, n(AB) = 2, n(A) = 14, and n(A' U B') = 47 The number of elements in region x is Z IS and the number of elements in region w is (Type whole numbers.) A W the number of elements in region y is m OU the number of elements in

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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**Using Venn Diagrams to Determine Set Elements** This lesson is focused on using Venn diagrams to quantify elements in different regions when provided with certain set operations. **Given Information:** - \( n(A \cup B) = 26 \): The number of elements in the union of sets A and B is 26. - \( n(A \cap B) = 2 \): The number of elements in the intersection of sets A and B is 2. - \( n(A) = 14 \): The number of elements in set A is 14. - \( n(A' \cup B') = 47 \): The number of elements in the union of the complements of sets A and B is 47. **Venn Diagram Explanation:** The provided image includes a Venn diagram with two overlapping circles representing sets A and B within a universal set U. The regions formed by the intersection and union are labeled as follows: - \( x \): The part of A that is not in B - \( y \): The part of B that is not in A - \( z \): The part of both A and B (intersection) - \( w \): The part outside both A and B within the universal set \( U \) **Determine the Elements:** The numbers for each region can be calculated using the given set information: 1. **Region \( x \):** Elements only in A (\( n(A) - n(A \cap B) = 14 - 2 = 12 \)) 2. **Region \( z \):** Elements in the intersection (\( n(A \cap B) = 2 \)) 3. **Region \( y \):** Elements only in B can be found by finding \( (n(A \cup B) - n(A) + n(A \cap B)) = 26 - 14 + 2 = 14 \) for total of B, then subtracting the intersection, \( 14 - 2 = 12 \). 4. **Region \( w \):** Elements in the universal set but not in A or B. The total elements in U is all elements in sets plus their complements, \( n(U) = n(A \cup B) + n(A' \cup B') = 26 + 47 = 73 \). Calculate: \( 73 - (x + y + z) = 73 - (12 +
Transcribed Image Text:**Using Venn Diagrams to Determine Set Elements** This lesson is focused on using Venn diagrams to quantify elements in different regions when provided with certain set operations. **Given Information:** - \( n(A \cup B) = 26 \): The number of elements in the union of sets A and B is 26. - \( n(A \cap B) = 2 \): The number of elements in the intersection of sets A and B is 2. - \( n(A) = 14 \): The number of elements in set A is 14. - \( n(A' \cup B') = 47 \): The number of elements in the union of the complements of sets A and B is 47. **Venn Diagram Explanation:** The provided image includes a Venn diagram with two overlapping circles representing sets A and B within a universal set U. The regions formed by the intersection and union are labeled as follows: - \( x \): The part of A that is not in B - \( y \): The part of B that is not in A - \( z \): The part of both A and B (intersection) - \( w \): The part outside both A and B within the universal set \( U \) **Determine the Elements:** The numbers for each region can be calculated using the given set information: 1. **Region \( x \):** Elements only in A (\( n(A) - n(A \cap B) = 14 - 2 = 12 \)) 2. **Region \( z \):** Elements in the intersection (\( n(A \cap B) = 2 \)) 3. **Region \( y \):** Elements only in B can be found by finding \( (n(A \cup B) - n(A) + n(A \cap B)) = 26 - 14 + 2 = 14 \) for total of B, then subtracting the intersection, \( 14 - 2 = 12 \). 4. **Region \( w \):** Elements in the universal set but not in A or B. The total elements in U is all elements in sets plus their complements, \( n(U) = n(A \cup B) + n(A' \cup B') = 26 + 47 = 73 \). Calculate: \( 73 - (x + y + z) = 73 - (12 +
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