To solve [tex]\( (A \cap B) \cup C \)[/tex] given the sets [tex]\( A = \{4, 5, 6, 7, 8, 9\} \)[/tex], [tex]\( B = \{1, 3, 5, 7, 9, 11, 13, 15\} \)[/tex], and [tex]\( C = \{2, 7, 9, 10, 12, 14\} \)[/tex], let's proceed with the following steps:
1. Find the Intersection of Sets A and B ([tex]\( A \cap B \)[/tex]):
- The intersection of two sets contains only the elements that are present in both sets.
- Therefore, find the common elements between [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
[tex]\[
A \cap B = \{4, 5, 6, 7, 8, 9\} \cap \{1, 3, 5, 7, 9, 11, 13, 15\} = \{5, 7, 9\}
\][/tex]
- Thus, [tex]\( A \cap B = \{5, 7, 9\} \)[/tex].
2. Find the Union of [tex]\( (A \cap B) \)[/tex] with Set C ([tex]\( (A \cap B) \cup C \)[/tex]):
- The union of two sets contains all elements that are in either set or both sets.
- Combine all elements of [tex]\( A \cap B \)[/tex] and [tex]\( C \)[/tex].
[tex]\[
(A \cap B) \cup C = \{5, 7, 9\} \cup \{2, 7, 9, 10, 12, 14\}
\][/tex]
- Combine the elements from both sets and remove any duplicates:
[tex]\[
(A \cap B) \cup C = \{2, 5, 7, 9, 10, 12, 14\}
\][/tex]
Thus, the final result is:
[tex]\[
(A \cap B) \cup C = \{2, 5, 7, 9, 10, 12, 14\}
\][/tex]
Additionally, the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is:
[tex]\[
A \cap B = \{5, 7, 9\}
\][/tex]
