The problem consists of two parts. Part a asks to find: i. The complement of set $B$, denoted as $\overline{B}$. ii. The intersection of the complement of set $A$ and the complement of set $B$, denoted as $\overline{A} \cap \overline{B}$. iii. The union of set $B$ and the complement of set $C$, denoted as $B \cup \overline{C}$. Part b provides the following information: $n(A) = 6$ $n(A \cap B) = 2$ $n(A \cup B) = 9$ The problem asks to find $n(B)$, the number of elements in set $B$.

1. Problem Description

The problem consists of two parts.

Part a asks to find:

i. The complement of set $B$, denoted as $\overline{B}$.

ii. The intersection of the complement of set $A$ and the complement of set $B$, denoted as $\overline{A} \cap \overline{B}$.

iii. The union of set $B$ and the complement of set $C$, denoted as $B \cup \overline{C}$.

Part b provides the following information:

$n(A) = 6$

$n(A \cap B) = 2$

$n(A \cup B) = 9$

The problem asks to find $n(B)$, the number of elements in set $B$.

2. Solution Steps

a.

i. The complement of a set $B$, denoted by $\overline{B}$, is the set of all elements in the universal set $U$ that are not in $B$. Given $U = \{a, b, c, d, e, f, g, h, i, j\}$ and $B = \{b, c, f, g\}$, then $\overline{B} = \{a, d, e, h, i, j\}$.

ii. We first find $\overline{A}$. Given $U = \{a, b, c, d, e, f, g, h, i, j\}$ and $A = \{a, c, f, g\}$, then $\overline{A} = \{b, d, e, h, i, j\}$. Now we find the intersection of $\overline{A}$ and $\overline{B}$, which is the set of all elements that are in both $\overline{A}$ and $\overline{B}$. $\overline{A} = \{b, d, e, h, i, j\}$ and $\overline{B} = \{a, d, e, h, i, j\}$. Thus, $\overline{A} \cap \overline{B} = \{d, e, h, i, j\}$.

iii. We first find $\overline{C}$. Given $U = \{a, b, c, d, e, f, g, h, i, j\}$ and $C = \{d, e, f\}$, then $\overline{C} = \{a, b, c, g, h, i, j\}$. Now we find the union of $B$ and $\overline{C}$, which is the set of all elements that are in either $B$ or $\overline{C}$ or both. $B = \{b, c, f, g\}$ and $\overline{C} = \{a, b, c, g, h, i, j\}$. Thus, $B \cup \overline{C} = \{a, b, c, f, g, h, i, j\}$.

b.

We use the formula for the number of elements in the union of two sets:

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$

We are given $n(A) = 6$, $n(A \cap B) = 2$, and $n(A \cup B) = 9$. We want to find $n(B)$.

Substituting the given values into the formula:

$9 = 6 + n(B) - 2$

$9 = 4 + n(B)$

$n(B) = 9 - 4$

$n(B) = 5$

3. Final Answer

a.

i. $\overline{B} = \{a, d, e, h, i, j\}$

ii. $\overline{A} \cap \overline{B} = \{d, e, h, i, j\}$

iii. $B \cup \overline{C} = \{a, b, c, f, g, h, i, j\}$

b.

$n(B) = 5$