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Let n(A) = 5 and n(AUB) = 8. (a) What are the possible values of n(B)? (b) If An B=Ø, what is the only possible value of n(B)? (a) The possible value(s) for n(B) is/are (Use a comma to separate answers as needed. Use ascending order.) (b) The only possible value for n(B), when An B=Øis

Let n(A) = 5 and n(AUB) = 8. (a) What are the possible values of n(B)? (b) If An B=Ø, what is the only possible value of n(B)? (a) The possible value(s) for n(B) is/are (Use a comma to separate answers as needed. Use ascending order.) (b) The only possible value for n(B), when An B=Øis

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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## Problem Statement Given: - \( n(A) = 5 \) - \( n(A \cup B) = 8 \) (a) What are the possible values of \( n(B) \)? (b) If \( A \cap B = \varnothing \), what is the only possible value of \( n(B) \)? --- ### Solution (a) The possible value(s) for \( n(B) \) is/are: \[ \boxed{3, 4, 5, 6, 7, 8} \] (Use a comma to separate answers as needed. Use ascending order.) (b) The only possible value for \( n(B) \), when \( A \cap B = \varnothing \), is: \[ \boxed{3} \] --- **Explanation:** 1. **Total Elements in \( A \cup B \)**: - Given \( n(A \cup B) = 8 \), this means the total number of elements in either set \( A \) or set \( B \) (or both) is 8. 2. **Elements in \( A \)**: - Given \( n(A) = 5 \), this means set \( A \) contains 5 elements. 3. **Possible Values of \( n(B) \)**: - The number of elements in \( B \) could range from including none of the elements of \( A \) (disjoint sets) to including all elements of \( A \) plus others. Hence, the possible values of \( n(B) \) are 3, 4, 5, 6, 7, or 8. 4. **Disjoint Sets Condition (\( A \cap B = \varnothing \))**: - If \( A \) and \( B \) are disjoint (have no overlap), the total number of elements in \( A \cup B \) is the sum of the number of elements in \( A \) and \( B \). Therefore, \( n(A) + n(B) = 8 \). - Given \( n(A) = 5 \), we calculate \( n(B) = 8 - 5 = 3 \). This exercise demonstrates the principles of set theory, specifically the relationships between union, intersection, and cardinality of sets.
Transcribed Image Text:## Problem Statement Given: - \( n(A) = 5 \) - \( n(A \cup B) = 8 \) (a) What are the possible values of \( n(B) \)? (b) If \( A \cap B = \varnothing \), what is the only possible value of \( n(B) \)? --- ### Solution (a) The possible value(s) for \( n(B) \) is/are: \[ \boxed{3, 4, 5, 6, 7, 8} \] (Use a comma to separate answers as needed. Use ascending order.) (b) The only possible value for \( n(B) \), when \( A \cap B = \varnothing \), is: \[ \boxed{3} \] --- **Explanation:** 1. **Total Elements in \( A \cup B \)**: - Given \( n(A \cup B) = 8 \), this means the total number of elements in either set \( A \) or set \( B \) (or both) is 8. 2. **Elements in \( A \)**: - Given \( n(A) = 5 \), this means set \( A \) contains 5 elements. 3. **Possible Values of \( n(B) \)**: - The number of elements in \( B \) could range from including none of the elements of \( A \) (disjoint sets) to including all elements of \( A \) plus others. Hence, the possible values of \( n(B) \) are 3, 4, 5, 6, 7, or 8. 4. **Disjoint Sets Condition (\( A \cap B = \varnothing \))**: - If \( A \) and \( B \) are disjoint (have no overlap), the total number of elements in \( A \cup B \) is the sum of the number of elements in \( A \) and \( B \). Therefore, \( n(A) + n(B) = 8 \). - Given \( n(A) = 5 \), we calculate \( n(B) = 8 - 5 = 3 \). This exercise demonstrates the principles of set theory, specifically the relationships between union, intersection, and cardinality of sets.
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