For the universal set, 

=*U* 2,  3,  4,  5,  6,  7 

, complete the parts below.
Write your answers in roster form or as 

∅

.



(a) Suppose 

=*B* 2,  5,  7 

. Then what is 

*B*′

?

*B*′

=

(b) Suppose we know that 

=*C*′ 2,  5,  6,  7 

. Then what would 

*C*

 have to be?

*C*

=

Given the **universal set** \( U = \{2, 3, 4, 5, 6, 7\} \), let's solve each part one by one.

### Part (a)

Suppose **\( B = \{2, 5, 7\} \)**. We need to find **\( B' \)**, which represents the complement of \( B \) in the universal set \( U \). The complement consists of all the elements in \( U \) that are **not** in \( B \).

- \( U = \{2, 3, 4, 5, 6, 7\} \)
- \( B = \{2, 5, 7\} \)

So, \( B' \) will be the elements in \( U \) but not in \( B \), which are:

\[
B' = \{3, 4, 6\}
\]

### Part (b)

We are given **\( C' = \{2, 5, 6, 7\} \)**, which represents the complement of set \( C \) in the universal set \( U \). We need to find \( C \).

Since \( C' \) consists of all the elements **not in \( C \)**, \( C \) will be the elements that are **in \( U \) but not in \( C' \)**.

- \( U = \{2, 3, 4, 5, 6, 7\} \)
- \( C' = \{2, 5, 6, 7\} \)

So, \( C \) will be the elements in \( U \) but not in \( C' \), which are:

\[
C = \{3, 4\}
\]

### Final Answers:

(a) \( B' = \{3, 4, 6\} \)

(b) \( C = \{3, 4\} \)

---

Would you like any more details or clarifications?

Here are 5 related questions to deepen your understanding:

1. What is the complement of a set, and how is it calculated?
2. If \( U = \{1, 2, 3, 4, 5\} \) and \( D = \{1, 2\} \), what is \( D' \)?
3. Can the complement of a set ever be the empty set \( \emptyset \)? Under what condition?
4. If \( A' = \{1, 3\} \) and \( U = \{1, 2, 3, 4\} \), what is \( A \)?
5. How is the concept of a universal set used in probability theory?

**Tip:** Always verify that the union of a set and its complement equals the universal set!

For the universal set U = {2, 3, 4, 5, 6, 7}, complete the parts below. (a) Suppose B = {2, 5, 7}. Then what is B'? (b) Suppose we know that C' = {2, 5, 6, 7}. Then what would C have to be?

Learn how to find the complement of sets B and C within the universal set U = {2, 3, 4, 5, 6, 7}. This problem covers key set theory concepts, including complement of sets, universal sets, and their mathematical properties. Suitable for students in Grades 6-8.

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