Use the given sets to find [tex]$(A \cap B) \cup C$[/tex].

[tex]\[
\begin{array}{l}
A=\{3, 4, 5, 6, 7, 8\} \\
B=\{1, 3, 5, 7, 9, 11, 13, 15\} \\
C=\{3, 4, 6, 10, 12, 14\}
\end{array}
\][/tex]



Answer :

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]