Certainly! Let's break down the steps to find [tex]\((A \cap B) \cup C\)[/tex] given the sets provided.
1. Define the sets:
- [tex]\( A = \{3, 4, 5, 6, 7, 8\} \)[/tex]
- [tex]\( B = \{1, 3, 5, 7, 9, 11, 13, 15\} \)[/tex]
- [tex]\( C = \{3, 4, 6, 10, 12, 14\} \)[/tex]
2. Find the intersection of sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
- The intersection [tex]\(A \cap B\)[/tex] consists of elements that are common in both sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
Let's list the elements of [tex]\(A\)[/tex] and [tex]\(B\)[/tex] to see which ones they have in common:
- [tex]\(A\)[/tex] contains: [tex]\(3, 4, 5, 6, 7, 8\)[/tex]
- [tex]\(B\)[/tex] contains: [tex]\(1, 3, 5, 7, 9, 11, 13, 15\)[/tex]
The common elements (intersection) are: [tex]\(3, 5, 7\)[/tex].
So, [tex]\(A \cap B = \{3, 5, 7\}\)[/tex].
3. Find the union of [tex]\( (A \cap B) \)[/tex] with set [tex]\(C\)[/tex]:
- The union of two sets consists of all unique elements from both sets.
Let's list the elements of [tex]\(A \cap B\)[/tex] and [tex]\(C\)[/tex] and combine them:
- [tex]\(A \cap B\)[/tex] contains: [tex]\(3, 5, 7\)[/tex]
- [tex]\(C\)[/tex] contains: [tex]\(3, 4, 6, 10, 12, 14\)[/tex]
Combining these, we get the union of [tex]\( (A \cap B) \)[/tex] and [tex]\(C\)[/tex]:
- [tex]\( \{3, 5, 7\} \cup \{3, 4, 6, 10, 12, 14\} \)[/tex]
Note that the element "3" appears in both sets, so we list it only once in the union.
Therefore, the union is: [tex]\( \{3, 4, 5, 6, 7, 10, 12, 14\} \)[/tex].
In conclusion:
- [tex]\(A \cap B = \{3, 5, 7\}\)[/tex]
- [tex]\((A \cap B) \cup C = \{3, 4, 5, 6, 7, 10, 12, 14\}\)[/tex]
So, the final result is:
[tex]\[
(A \cap B) \cup C = \{3, 4, 5, 6, 7, 10, 12, 14\}
\][/tex]
