Sure, let's solve the problem step by step:
Given sets:
[tex]\[ A = \{4, 5, 6, 7, 8, 9\} \][/tex]
[tex]\[ B = \{2, 4, 6, 10, 12, 14\} \][/tex]
[tex]\[ C = \{1, 3, 5, 7, 9, 11, 13, 15\} \][/tex]
### Step 1: Find the union of sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex]
The union of two sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex], denoted [tex]\(A \cup B\)[/tex], is the set of elements that are in [tex]\(A\)[/tex], in [tex]\(B\)[/tex], or in both. To find the union, we combine all the unique elements from both sets.
[tex]\[
A \cup B = \{2, 4, 5, 6, 7, 8, 9, 10, 12, 14\}
\][/tex]
### Step 2: Find the intersection of the union [tex]\((A \cup B)\)[/tex] with set [tex]\(C\)[/tex]
The intersection of sets [tex]\(X\)[/tex] and [tex]\(Y\)[/tex], denoted [tex]\(X \cap Y\)[/tex], is the set of elements that are in both [tex]\(X\)[/tex] and [tex]\(Y\)[/tex]. To find the intersection of [tex]\(A \cup B\)[/tex] with [tex]\(C\)[/tex], we need to find the common elements between the set [tex]\(A \cup B\)[/tex] and the set [tex]\(C\)[/tex].
[tex]\[
A \cup B = \{2, 4, 5, 6, 7, 8, 9, 10, 12, 14\}
\][/tex]
[tex]\[
C = \{1, 3, 5, 7, 9, 11, 13, 15\}
\][/tex]
The common elements between them are [tex]\(5, 7,\)[/tex] and [tex]\(9\)[/tex].
Thus,
[tex]\[
(A \cup B) \cap C = \{5, 7, 9\}
\][/tex]
### Conclusion
The set [tex]\((A \cup B) \cap C\)[/tex] is:
[tex]\[
(A \cup B) \cap C = \{5, 7, 9\}
\][/tex]
