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Given [tex]\(A=\{1,2,3\}\)[/tex], [tex]\(B=\{2,4,6\}\)[/tex], and [tex]\(C=\{1,2,3,4,5,6\}\)[/tex], find [tex]\(A \cap (B \cap C)\)[/tex].

A. [tex]\(\emptyset\)[/tex]

B. [tex]\(\{1,2,3,4,5,6,7,8\}\)[/tex]

C. [tex]\(\{2\}\)[/tex]



Answer :

GinnyAnswer
Step-by-Step Solution:

1. Identify the sets:
- Given the sets:
- [tex]\(A = \{1, 2, 3\}\)[/tex]
- [tex]\(B = \{2, 4, 6\}\)[/tex]
- [tex]\(C = \{1, 2, 3, 4, 5, 6\}\)[/tex]

2. Find the intersection of [tex]\(B\)[/tex] and [tex]\(C\)[/tex]:
- [tex]\(B \cap C\)[/tex] is the set of elements that are common to both [tex]\(B\)[/tex] and [tex]\(C\)[/tex].
- That means we look for the common elements in [tex]\(\{2, 4, 6\}\)[/tex] and [tex]\(\{1, 2, 3, 4, 5, 6\}\)[/tex].
- The common elements are [tex]\(\{2, 4, 6\}\)[/tex].

Thus, [tex]\(B \cap C = \{2, 4, 6\}\)[/tex].

3. Find the intersection of [tex]\(A\)[/tex] and [tex]\(B \cap C\)[/tex]:
- [tex]\(A \cap (B \cap C)\)[/tex] is the set of elements that are common to both [tex]\(A\)[/tex] and [tex]\(B \cap C\)[/tex].
- So, we need to find the common elements in [tex]\(\{1, 2, 3\}\)[/tex] and [tex]\(\{2, 4, 6\}\)[/tex].
- The only common element is [tex]\(\{2\}\)[/tex].

Hence, [tex]\(A \cap (B \cap C) = \{2\}\)[/tex].

4. Conclusion:
- The result of [tex]\(A \cap (B \cap C)\)[/tex] is [tex]\(\{2\}\)[/tex].

Therefore, [tex]\(A \cap (B \cap C) = \{2\} \)[/tex].

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