To solve this problem, let's break it down step by step:
1. Understand the sets and elements:
You are given that the Cartesian product of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] has 6 elements and includes the elements [tex]\((3,2),(7,5),\)[/tex] and [tex]\((8,5)\)[/tex].
2. Determine elements in sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
From the given elements of the Cartesian product [tex]\((3,2),(7,5),\)[/tex] and [tex]\((8,5)\)[/tex], we can observe:
- The first coordinates [tex]\(3, 7,\)[/tex] and [tex]\(8\)[/tex] belong to set [tex]\(A\)[/tex].
- The second coordinates [tex]\(2\)[/tex] and [tex]\(5\)[/tex] belong to set [tex]\(B\)[/tex].
Hence, it can be inferred that:
[tex]\[
A = \{3, 7, 8\}
\][/tex]
[tex]\[
B = \{2, 5\}
\][/tex]
3. Verify the Cartesian product size:
Verify that the Cartesian product [tex]\(A \times B\)[/tex] contains 6 elements.
[tex]\[
n(A) = 3 \quad (\text{since } A = \{3, 7, 8\})
\][/tex]
[tex]\[
n(B) = 2 \quad (\text{since } B = \{2, 5\})
\][/tex]
The number of elements in the Cartesian product is:
[tex]\[
n(A \times B) = n(A) \cdot n(B) = 3 \cdot 2 = 6
\][/tex]
This is consistent with the problem statement.
4. Assess each option:
- Option a: [tex]\( A = \{3, 7, 8\} \)[/tex] — This is correct.
- Option b: [tex]\( B = \{2, 5, 7\} \)[/tex] — This is incorrect since set [tex]\(B\)[/tex] only contains [tex]\(2\)[/tex] and [tex]\(5\)[/tex].
- Option c: [tex]\( B = \{3, 5\} \)[/tex] — This is incorrect because set [tex]\(B\)[/tex] does not contain [tex]\(3\)[/tex].
- Option d: None of these — This is incorrect because option (a) is correct.
Therefore, the correct answer is:
[tex]\[
\boxed{a. \, A = \{3, 7, 8\}}
\][/tex]
