Question
For the universal set, \( U=\{f, g, h, p, q, r, x\} \), complete the parts below. Write your answers in roster form or as \( \varnothing \). (a) Suppose \( A=\{f, g, q, r, x\} \). Then what is \( A^{\prime} \) ? \( A^{\prime}=\square \) (b) Suppose we know that \( B^{\prime}=\{p, q, r\} \). Then what would \( B \) have to be? \( B=\square \)
Ask by Norton Cox. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( A^{\prime} = \{h, p\} \)
\( B = \{f, g, h, x\} \)
Solution
Let's solve each part step by step.
### Part (a)
**Given:**
- Universal set \( U = \{f, g, h, p, q, r, x\} \)
- Set \( A = \{f, g, q, r, x\} \)
**Find \( A^{\prime} \) (the complement of \( A \)):**
The complement of \( A \) consists of all elements in \( U \) that are not in \( A \).
\[
A^{\prime} = U \setminus A = \{h, p\}
\]
### Part (b)
**Given:**
- \( B^{\prime} = \{p, q, r\} \)
**Find \( B \):**
Since \( B^{\prime} \) is the complement of \( B \), \( B \) consists of all elements in \( U \) that are not in \( B^{\prime} \).
\[
B = U \setminus B^{\prime} = \{f, g, h, x\}
\]
### Final Answers
(a) \( A^{\prime} = \{h, p\} \)
(b) \( B = \{f,\, g,\, h,\, x\} \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To find \( A^{\prime} \), we need to identify the elements in the universal set \( U \) that are not in set \( A \). Since \( A=\{f, g, q, r, x\} \), the elements in \( A^{\prime} \) would be those from \( U \) that are not included in \( A \). (a) Thus, the elements of \( U \) that are not in \( A \) are \( \{h, p\} \). So, \( A^{\prime} = \{h, p\} \). (b) Since we know \( B^{\prime} = \{p, q, r\} \) is the complement of \( B \), this means that \( B \) will include all elements in \( U \) not in \( B^{\prime} \). The universal set \( U \) minus elements of \( B^{\prime} \) results in \( B = \{f, g, h, x\} \).
