Sure, let's go through the question step-by-step to find [tex]\((A \cup B) \cup C\)[/tex].
1. Define the sets:
- [tex]\(A = \{1, 2, 3, 4, 5, 6, 7, 8\}\)[/tex]
- [tex]\(B = \{2, 4, 8, 10, 12, 14\}\)[/tex]
- [tex]\(C = \{5, 7, 9, 11, 13, 15\}\)[/tex]
2. First, find the union of sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
The union of two sets includes all the elements from both sets, without duplicates.
[tex]\[
A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8\} \cup \{2, 4, 8, 10, 12, 14\}
\][/tex]
Since the union contains all unique elements from both sets:
[tex]\[
A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14\}
\][/tex]
3. Next, find the union of [tex]\((A \cup B)\)[/tex] and [tex]\(C\)[/tex]:
Similarly, the union of [tex]\((A \cup B)\)[/tex] and [tex]\(C\)[/tex] will include all unique elements from both [tex]\((A \cup B)\)[/tex] and [tex]\(C\)[/tex].
[tex]\[
(A \cup B) \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14\} \cup \{5, 7, 9, 11, 13, 15\}
\][/tex]
Including all unique elements:
[tex]\[
(A \cup B) \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}
\][/tex]
4. Final results:
- [tex]\(\mathbf{A \cup B}\)[/tex] results in [tex]\(\{1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14\}\)[/tex]
- [tex]\(\mathbf{(A \cup B) \cup C}\)[/tex] results in [tex]\(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\)[/tex]
Therefore, the final answer for [tex]\((A \cup B) \cup C\)[/tex] is:
[tex]\[
\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}
\][/tex]
