Let's solve the given problem step-by-step involving the operations on sets [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex].
First, we need to clearly define the sets:
- [tex]\( A = \{ \text{multiples of 3 less than 12} \} = \{3, 6, 9\} \)[/tex]
- [tex]\( B = \{ \text{integers between 4 and 8} \} = \{5, 6, 7\} \)[/tex]
- [tex]\( C = \{4, 5, 7\} \)[/tex]
With these sets defined, let's proceed with the parts of the problem.
(i) [tex]\( A \cap B \)[/tex]
The intersection of two sets includes all elements that are present in both sets. Here we need to find the common elements between [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
- [tex]\(A = \{3, 6, 9\}\)[/tex]
- [tex]\(B = \{5, 6, 7\}\)[/tex]
The common element between these sets is [tex]\(6\)[/tex].
So,
[tex]\[ A \cap B = \{6\} \][/tex]
(ii) [tex]\( (A \cup B) \cap C \)[/tex]
First, we need to find the union of sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex], then we will find the intersection of the resulting set with [tex]\(C\)[/tex].
- [tex]\(A = \{3, 6, 9\}\)[/tex]
- [tex]\(B = \{5, 6, 7\}\)[/tex]
The union of [tex]\(A\)[/tex] and [tex]\(B\)[/tex] is all elements that are in either [tex]\(A\)[/tex] or [tex]\(B\)[/tex]:
[tex]\[ A \cup B = \{3, 6, 9, 5, 7\} \][/tex]
Next, we find the intersection of this union with [tex]\(C\)[/tex]:
- [tex]\(A \cup B = \{3, 6, 9, 5, 7\}\)[/tex]
- [tex]\(C = \{4, 5, 7\}\)[/tex]
The common elements here are [tex]\(5\)[/tex] and [tex]\(7\)[/tex].
So,
[tex]\[ (A \cup B) \cap C = \{5, 7\} \][/tex]
(iii) [tex]\( (A \cap B) \cup C \)[/tex]
First, we already found the intersection [tex]\(A \cap B\)[/tex] in part (i):
[tex]\[ A \cap B = \{6\} \][/tex]
Next, we form the union of this intersection with set [tex]\(C\)[/tex]:
- [tex]\(A \cap B = \{6\}\)[/tex]
- [tex]\(C = \{4, 5, 7\}\)[/tex]
The union is all elements that are in either [tex]\(\{6\}\)[/tex] or [tex]\(\{4, 5, 7\}\)[/tex]:
[tex]\[ (A \cap B) \cup C = \{4, 5, 6, 7\} \][/tex]
Finally, let's summarize the results:
(i) [tex]\( A \cap B = \{6\} \)[/tex]
(ii) [tex]\( (A \cup B) \cap C = \{5, 7\} \)[/tex]
(iii) [tex]\( (A \cap B) \cup C = \{4, 5, 6, 7\} \)[/tex]
