Question
For the universal set, \( U=\{4,5,6,7,8,9\} \), complete the parts below. Write your answers in roster form or as \( \varnothing \). (a) Suppose we know that \( A^{\prime}=\{4,6\} \). Then what would \( A \) have to be? \[ A=\square \] (b) Suppose \( B=\{5,6,7,8,9\} \). Then what is \( B^{\prime} \) ? \[ B^{\prime}=\square \]
Ask by Hampton Parsons. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( A = \{5,7,8,9\} \)
\( B' = \{4\} \)
Solution
Let's solve each part step by step.
**Given:**
- Universal set \( U = \{4,5,6,7,8,9\} \)
### (a) Finding \( A \) when \( A' = \{4,6\} \)
- \( A' \) represents the complement of \( A \) with respect to \( U \).
- To find \( A \), subtract \( A' \) from \( U \):
\[
A = U \setminus A' = \{4,5,6,7,8,9\} \setminus \{4,6\} = \{5,7,8,9\}
\]
**Answer:**
\[
A = \{5,7,8,9\}
\]
### (b) Finding \( B' \) when \( B = \{5,6,7,8,9\} \)
- \( B' \) represents the complement of \( B \) with respect to \( U \).
- To find \( B' \), subtract \( B \) from \( U \):
\[
B' = U \setminus B = \{4,5,6,7,8,9\} \setminus \{5,6,7,8,9\} = \{4\}
\]
**Answer:**
\[
B' = \{4\}
\]
---
**Final Answers:**
(a) \( A = \{5,7,8,9\} \)
(b) \( B^{\prime} = \{4\} \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
Given that \( A^{\prime}=\{4,6\} \), this means \( A \) contains all elements of the universal set \( U \) that are not in \( A^{\prime} \). Thus, we can identify that: \[ A = \{5,7,8,9\} \] For set \( B \), since \( B=\{5,6,7,8,9\} \), we find \( B^{\prime} \) by looking for the elements in the universal set \( U \) that are not in \( B \). The only element not included is: \[ B^{\prime}=\{4\} \]
