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
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a817796-b4be-478d-ba8b-56db553c444f%2F6e26b098-e57e-4807-a5bc-c7a462f64b82%2Fs56tf8m_processed.png&w=3840&q=75)
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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