Sure, let's go through each part of the problem step-by-step.
### Part (a) Show that \( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \)
1. Identify Sets:
[tex]\[
\begin{aligned}
&A = \{x \in \mathbb{N} : x \leq 7\} = \{1, 2, 3, 4, 5, 6, 7\} \\
&B = \{2, 3, 5, 7\} \\
&C = \{1, 3, 5, 7, 9\}
\end{aligned}
\][/tex]
2. Calculate \( B \cap C \):
[tex]\[
B \cap C = \{2, 3, 5, 7\} \cap \{1, 3, 5, 7, 9\} = \{3, 5, 7\}
\][/tex]
3. Calculate \( A \cup (B \cap C) \):
[tex]\[
A \cup (B \cap C) = \{1, 2, 3, 4, 5, 6, 7\} \cup \{3, 5, 7\} = \{1, 2, 3, 4, 5, 6, 7\}
\][/tex]
4. Calculate \( A \cup B \):
[tex]\[
A \cup B = \{1, 2, 3, 4, 5, 6, 7\} \cup \{2, 3, 5, 7\} = \{1, 2, 3, 4, 5, 6, 7\}
\][/tex]
5. Calculate \( A \cup C \):
[tex]\[
A \cup C = \{1, 2, 3, 4, 5, 6, 7\} \cup \{1, 3, 5, 7, 9\} = \{1, 2, 3, 4, 5, 6, 7, 9\}
\][/tex]
6. Calculate \( (A \cup B) \cap (A \cup C) \):
[tex]\[
(A \cup B) \cap (A \cup C) = \{1, 2, 3, 4, 5, 6, 7\} \cap \{1, 2, 3, 4, 5, 6, 7, 9\} = \{1, 2, 3, 4, 5, 6, 7\}
\][/tex]
7. Comparison:
[tex]\[
A \cup (B \cap C) = \{1, 2, 3, 4, 5, 6, 7\}
\][/tex]
[tex]\[
(A \cup B) \cap (A \cup C) = \{1, 2, 3, 4, 5, 6, 7\}
\][/tex]
Thus, we have shown:
[tex]\[
A \cup (B \cap C) = (A \cup B) \cap (A \cup C)
\][/tex]
### Part (b) Show that \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
1. Calculate \( B \cup C \):
[tex]\[
B \cup C = \{2, 3, 5, 7\} \cup \{1, 3, 5, 7, 9\} = \{1, 2, 3, 5, 7, 9\}
\][/tex]
2. Calculate \( A \cap (B \cup C) \):
[tex]\[
A \cap (B \cup C) = \{1, 2, 3, 4, 5, 6, 7\} \cap \{1, 2, 3, 5, 7, 9\} = \{1, 2, 3, 5, 7\}
\][/tex]
3. Calculate \( A \cap B \):
[tex]\[
A \cap B = \{1, 2, 3, 4, 5, 6, 7\} \cap \{2, 3, 5, 7\} = \{2, 3, 5, 7\}
\][/tex]
4. Calculate \( A \cap C \):
[tex]\[
A \cap C = \{1, 2, 3, 4, 5, 6, 7\} \cap \{1, 3, 5, 7, 9\} = \{1, 3, 5, 7\}
\][/tex]
5. Calculate \( (A \cap B) \cup (A \cap C) \):
[tex]\[
(A \cap B) \cup (A \cap C) = \{2, 3, 5, 7\} \cup \{1, 3, 5, 7\} = \{1, 2, 3, 5, 7\}
\][/tex]
6. Comparison:
[tex]\[
A \cap (B \cup C) = \{1, 2, 3, 5, 7\}
\][/tex]
[tex]\[
(A \cap B) \cup (A \cap C) = \{1, 2, 3, 5, 7\}
\][/tex]
Thus, we have shown:
[tex]\[
A \cap (B \cup C) = (A \cap B) \cup (A \cap C)
\][/tex]
This completes the solution.
