Question
For the universal set, \( U=\{5,6,7,8,9\} \), complete the parts below. Write your answers in roster form or as \( \varnothing \). (a) Suppose \( A=\{5,7,8,9\} \). Then what is \( A^{\prime} \) ? \[ A^{\prime}=\square \] (b) Suppose we know that \( B^{\prime}=\{6,8\} \). Then what would \( B \) have to be? \[ B=\square \]
Ask by Wagner Mitchell. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
(a) \( A^{\prime} = \{6\} \)
(b) \( B = \{5, 7, 9\} \)
Solution
Let's solve each part step by step.
### Part (a)
**Given:**
- Universal set: \( U = \{5, 6, 7, 8, 9\} \)
- Set \( A = \{5, 7, 8, 9\} \)
**To find:**
- The complement of \( A \), denoted \( A^{\prime} \), which includes all elements in \( U \) that are not in \( A \).
**Solution:**
\[
A^{\prime} = U \setminus A = \{5, 6, 7, 8, 9\} \setminus \{5, 7, 8, 9\} = \{6\}
\]
**Answer:**
\[
A^{\prime} = \{6\}
\]
### Part (b)
**Given:**
- Universal set: \( U = \{5, 6, 7, 8, 9\} \)
- The complement of set \( B \), denoted \( B^{\prime} = \{6, 8\} \)
**To find:**
- The set \( B \), which includes all elements in \( U \) that are not in \( B^{\prime} \).
**Solution:**
\[
B = U \setminus B^{\prime} = \{5, 6, 7, 8, 9\} \setminus \{6, 8\} = \{5, 7, 9\}
\]
**Answer:**
\[
B = \{5, 7, 9\}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To find \( A^{\prime} \), which is the complement of set \( A \), we look for the elements in the universal set \( U \) that are not in \( A \). Given \( A=\{5,7,8,9\} \) and \( U=\{5,6,7,8,9\} \), we see that the only element from \( U \) not in \( A \) is 6. Therefore, \[ A^{\prime}=\{6\} \] For part (b), we know that \( B^{\prime}=\{6,8\} \) includes the elements not in set \( B \). The universal set \( U \) minus the elements of \( B^{\prime} \) will give us \( B \). From \( U=\{5,6,7,8,9\} \), removing \( 6 \) and \( 8 \), we have the elements \( 5, 7, 9 \) remaining in \( B \). Thus, \[ B=\{5,7,9\} \]
