Let's find the elements for each of the sets as specified:
1. Intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] ([tex]\( A \cap B \)[/tex]):
- We look for elements that are common in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
- [tex]\( A \cap B = \{3, 5, 6\} \)[/tex].
2. Union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] ([tex]\( A \cup B \)[/tex]):
- We combine all elements from sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], without duplicates.
- [tex]\( A \cup B = \{0, 1, 3, 5, 6, 8\} \)[/tex].
3. Intersection of the Union of [tex]\( B \)[/tex] and [tex]\( C \)[/tex] with [tex]\( A \)[/tex] [tex]\((B \cup C) \cap A\)[/tex]:
- First, find the union of [tex]\( B \)[/tex] and [tex]\( C \)[/tex]: [tex]\( B \cup C = \{0, 2, 3, 4, 5, 6, 7, 8\} \)[/tex].
- Then find the intersection of this union with [tex]\( A \)[/tex]:
- Elements in [tex]\( A \)[/tex]: \{0, 1, 3, 5, 6\}
- Intersection: \{0, 3, 5, 6\}
- Therefore, [tex]\((B \cup C) \cap A = \{0, 3, 5, 6\} \)[/tex].
4. Union of [tex]\( B \)[/tex] with the Intersection of [tex]\( C \)[/tex] and [tex]\( A \)[/tex] [tex]\(B \cup (C \cap A)\)[/tex]:
- First, find the intersection of [tex]\( C \)[/tex] and [tex]\( A \)[/tex]: [tex]\( C \cap A = \{0, 5\} \)[/tex].
- Then find the union of [tex]\( B \)[/tex] with this intersection:
- Elements in [tex]\( B \)[/tex]: \{3, 5, 6, 8\}
- Union: \{0, 3, 5, 6, 8\}
- Therefore, [tex]\( B \cup (C \cap A) = \{0, 3, 5, 6, 8\} \)[/tex].
Hence, the solutions for the given sets are:
[tex]\[ \begin{aligned}
&A \cap B = \{3, 5, 6\}, \\
&A \cup B = \{0, 1, 3, 5, 6, 8\}, \\
&(B \cup C) \cap A = \{0, 3, 5, 6\}, \\
&B \cup (C \cap A) = \{0, 3, 5, 6, 8\}.
\end{aligned} \][/tex]
