To find [tex]\(A \cup B\)[/tex], which is the union of the sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex], we need to combine all the unique elements from sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
Given sets:
[tex]\[ A = \{1, 3, 5, 7\} \][/tex]
[tex]\[ B = \{1, 2, 3, 4\} \][/tex]
The union of two sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex] includes each element that is in either [tex]\(A\)[/tex] or [tex]\(B\)[/tex], without duplication. Therefore, we need to list all the unique elements from both sets.
Step-by-step process:
1. Start with all elements from set [tex]\(A\)[/tex]:
[tex]\[ A = \{1, 3, 5, 7\} \][/tex]
2. Add all elements from set [tex]\(B\)[/tex], but only include those that are not already in [tex]\(A\)[/tex], to avoid duplication:
[tex]\[ B = \{1, 2, 3, 4\} \][/tex]
3. Combine the elements:
[tex]\[ A \cup B = \{1, 3, 5, 7\} \cup \{1, 2, 3, 4\} \][/tex]
4. List the unique elements from both sets:
[tex]\[ A \cup B = \{1, 2, 3, 4, 5, 7\} \][/tex]
Comparing this result with the provided options:
[tex]\[
\{1,2,3,4,5,7\} \\
\{2,3,4,6,7,10\} \\
\{1,3,6,7,8,10\} \\
\{1,2,3,5,8,10\} \\
\{2,3,5,7,8,9\} \\
\emptyset
\][/tex]
The correct choice that matches our result is:
[tex]\[ \{1, 2, 3, 4, 5, 7\} \][/tex]
