The problem defines a universal set $U$ and several subsets $A, B, C, D, E, F$. The task is to compute various set operations based on these sets. Specifically, we need to find: (a) $A \cup B$, $A \cap B$, $U^c$, $A \setminus B$, $B \setminus A$, $D \setminus E$, $F \setminus D$ (b) $A \Delta B$, $C \Delta D$, $E \Delta F$ (c) $A \cap (B \cup E)$, $(A \setminus B)^c$, $(A \cap D) \setminus B$, $(B \cap F) \cup (C \cap E)$, $B \times F$

1. Problem Description

The problem defines a universal set $U$ and several subsets $A, B, C, D, E, F$. The task is to compute various set operations based on these sets. Specifically, we need to find:

(a) $A \cup B$, $A \cap B$, $U^c$, $A \setminus B$, $B \setminus A$, $D \setminus E$, $F \setminus D$

(b) $A \Delta B$, $C \Delta D$, $E \Delta F$

(c) $A \cap (B \cup E)$, $(A \setminus B)^c$, $(A \cap D) \setminus B$, $(B \cap F) \cup (C \cap E)$, $B \times F$

2. Solution Steps

First, let's explicitly list the elements of each set:

$U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$

$A = \{1, 2, 3, 4, 5\}$

$B = \{4, 5, 6, 7\}$

$C = \{5, 6, 7, 8, 9\}$

$D = \{1, 3, 5, 7, 9\}$ (odd numbers in $U$)

$E = \{2, 4, 6, 8\}$ (even numbers)

$F = \{1, 5, 9\}$

(a)

$A \cup B = \{1, 2, 3, 4, 5, 6, 7\}$

$A \cap B = \{4, 5\}$

$U^c = \emptyset$ (complement of $U$ with respect to $U$ is empty)

$A \setminus B = \{1, 2, 3\}$

$B \setminus A = \{6, 7\}$

$D \setminus E = \{1, 3, 5, 7, 9\}$ ($D \setminus E = D$ since $D$ and $E$ are disjoint)

$F \setminus D = \emptyset$

(b)

$A \Delta B = (A \setminus B) \cup (B \setminus A) = \{1, 2, 3\} \cup \{6, 7\} = \{1, 2, 3, 6, 7\}$

$C \Delta D = (C \setminus D) \cup (D \setminus C) = \{6, 8\} \cup \{1, 3\} = \{1, 3, 6, 8\}$

$E \Delta F = (E \setminus F) \cup (F \setminus E) = \{2, 4, 6, 8\} \cup \{1, 5, 9\} = \{1, 2, 4, 5, 6, 8, 9\}$

(c)

$B \cup E = \{2, 4, 5, 6, 7, 8\}$

$A \cap (B \cup E) = \{1, 2, 3, 4, 5\} \cap \{2, 4, 5, 6, 7, 8\} = \{2, 4, 5\}$

$A \setminus B = \{1, 2, 3\}$

$(A \setminus B)^c = U \setminus (A \setminus B) = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \setminus \{1, 2, 3\} = \{4, 5, 6, 7, 8, 9\}$

$A \cap D = \{1, 2, 3, 4, 5\} \cap \{1, 3, 5, 7, 9\} = \{1, 3, 5\}$

$(A \cap D) \setminus B = \{1, 3, 5\} \setminus \{4, 5, 6, 7\} = \{1, 3\}$

$B \cap F = \{4, 5, 6, 7\} \cap \{1, 5, 9\} = \{5\}$

$C \cap E = \{5, 6, 7, 8, 9\} \cap \{2, 4, 6, 8\} = \{6, 8\}$

$(B \cap F) \cup (C \cap E) = \{5\} \cup \{6, 8\} = \{5, 6, 8\}$

$B \times F = \{(4,1), (4,5), (4,9), (5,1), (5,5), (5,9), (6,1), (6,5), (6,9), (7,1), (7,5), (7,9)\}$

3. Final Answer

(a) $A \cup B = \{1, 2, 3, 4, 5, 6, 7\}$, $A \cap B = \{4, 5\}$, $U^c = \emptyset$, $A \setminus B = \{1, 2, 3\}$, $B \setminus A = \{6, 7\}$, $D \setminus E = \{1, 3, 5, 7, 9\}$, $F \setminus D = \emptyset$

(b) $A \Delta B = \{1, 2, 3, 6, 7\}$, $C \Delta D = \{1, 3, 6, 8\}$, $E \Delta F = \{1, 2, 4, 5, 6, 8, 9\}$

(c) $A \cap (B \cup E) = \{2, 4, 5\}$, $(A \setminus B)^c = \{4, 5, 6, 7, 8, 9\}$, $(A \cap D) \setminus B = \{1, 3\}$, $(B \cap F) \cup (C \cap E) = \{5, 6, 8\}$, $B \times F = \{(4,1), (4,5), (4,9), (5,1), (5,5), (5,9), (6,1), (6,5), (6,9), (7,1), (7,5), (7,9)\}$