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) $C = \{3, 4\}$

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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!