Sure, let's go through this step by step.
### a. List the Elements of Set B and Set C
Given Set [tex]\( A = \{1, 2, 3, 4, 5\} \)[/tex]:
1. For Set B:
- Determined by the rule [tex]\( B = \{y: y = x + 3, x \in A\} \)[/tex].
- Compute the elements of Set B:
[tex]\[
\begin{align*}
\text{If } x = 1, & \quad y = 1 + 3 = 4 \\
\text{If } x = 2, & \quad y = 2 + 3 = 5 \\
\text{If } x = 3, & \quad y = 3 + 3 = 6 \\
\text{If } x = 4, & \quad y = 4 + 3 = 7 \\
\text{If } x = 5, & \quad y = 5 + 3 = 8 \\
\end{align*}
\][/tex]
- Therefore, Set B is [tex]\( \{4, 5, 6, 7, 8\} \)[/tex].
2. For Set C:
- Determined by the rule [tex]\( C = \{Z: Z = 2x, x \in A\} \)[/tex].
- Compute the elements of Set C:
[tex]\[
\begin{align*}
\text{If } x = 1, & \quad Z = 2 \times 1 = 2 \\
\text{If } x = 2, & \quad Z = 2 \times 2 = 4 \\
\text{If } x = 3, & \quad Z = 2 \times 3 = 6 \\
\text{If } x = 4, & \quad Z = 2 \times 4 = 8 \\
\text{If } x = 5, & \quad Z = 2 \times 5 = 10 \\
\end{align*}
\][/tex]
- Therefore, Set C is [tex]\( \{2, 4, 6, 8, 10\} \)[/tex].
### b. Type of Set B and Set C
- Both Set B and Set C are finite sets.
- A finite set is a set that has a limited number of elements. Since both Set B and Set C have a specific, countable number of elements, they are finite.
### c. Venn Diagram of Sets B and C
To draw the Venn diagram for Sets B (\{4, 5, 6, 7, 8\}) and C (\{2, 4, 6, 8, 10\}), you can follow these steps:
1. Overlap/Intersection: Identify the common elements:
[tex]\[
B \cap C = \{4, 6, 8\}
\][/tex]
2. Separate Parts:
- B only: \{5, 7\}
- C only: \{2, 10\}
Here is the Venn diagram:
```
______
/ \
/ B \
/ (4,6,8) \
/ 5,7 \
/_____________\
\ ______ /
\ / \
\6 / C \
\(4,6,8) \
\ 2,10 /
\_______/
```
In the Venn Diagram:
- The intersection (the overlapping area) of the two circles contains the elements \{4, 6, 8\}.
- The part exclusive to B contains \{5, 7\}.
- The part exclusive to C contains \{2, 10\}.
This concludes the step-by-step detailed solution for the given problem.
