Given the universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and the sets $A = \{1, 2, 3, 4, 5\}$, $B = \{2, 4, 6\}$, and $C = \{1, 3, 5\}$, we are asked to find: (1) $A \cup B$, the union of sets A and B. (2) $A \setminus B$, the set difference between A and B. (3) $A \cap C$, the intersection of sets A and C. (4) $C \times B$, the Cartesian product of sets C and B.

1. Problem Description

Given the universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and the sets $A = \{1, 2, 3, 4, 5\}$, $B = \{2, 4, 6\}$, and $C = \{1, 3, 5\}$, we are asked to find:

(1) $A \cup B$, the union of sets A and B.

(2) $A \setminus B$, the set difference between A and B.

(3) $A \cap C$, the intersection of sets A and C.

(4) $C \times B$, the Cartesian product of sets C and B.

2. Solution Steps

(1) Find $A \cup B$:

$A \cup B$ is the set of all elements that are in A or in B or in both.

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

$B = \{2, 4, 6\}$

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

(2) Find $A \setminus B$:

$A \setminus B$ is the set of all elements that are in A but not in B.

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

$B = \{2, 4, 6\}$

$A \setminus B = \{1, 3, 5\}$

(3) Find $A \cap C$:

$A \cap C$ is the set of all elements that are in both A and C.

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

$C = \{1, 3, 5\}$

$A \cap C = \{1, 3, 5\}$

(4) Find $C \times B$:

$C \times B$ is the set of all ordered pairs $(c, b)$ such that $c \in C$ and $b \in B$.

$C = \{1, 3, 5\}$

$B = \{2, 4, 6\}$

$C \times B = \{(1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6)\}$

3. Final Answer

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

(2) $A \setminus B = \{1, 3, 5\}$

(3) $A \cap C = \{1, 3, 5\}$

(4) $C \times B = \{(1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6)\}$