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Answer :
To solve the problem of finding the smallest set [tex]\(X\)[/tex] such that [tex]\(X \cup \{1, 2\} = \{1, 2, 3, 5, 7, 9\}\)[/tex], let's follow these steps:
1. Understand the Union: The expression [tex]\(X \cup \{1, 2\}\)[/tex] means combining all elements from set [tex]\(X\)[/tex] and the set [tex]\(\{1, 2\}\)[/tex] without repeating any element.
2. Identify the Full Set: We want the result of this union to be [tex]\(\{1, 2, 3, 5, 7, 9\}\)[/tex].
3. Determine Missing Elements: Start with the set [tex]\(\{1, 2, 3, 5, 7, 9\}\)[/tex] and remove the elements \{1, 2\}, because these are already part of both sets in the union:
- Remove 1 and 2 from \{1, 2, 3, 5, 7, 9\}.
4. Find Set [tex]\(X\)[/tex]: After removing \{1, 2\}, we are left with \{3, 5, 7, 9\}. These are the elements that must be in set [tex]\(X\)[/tex] to complete the union with \{1, 2\} and reach \{1, 2, 3, 5, 7, 9\}.
5. Identify the Smallest Set: Therefore, the smallest set [tex]\(X\)[/tex] that satisfies the equation is \{3, 5, 7, 9\}.
By following these steps, we determine that the correct choice is option (d) [tex]\(\{3, 5, 7, 9\}\)[/tex].
1. Understand the Union: The expression [tex]\(X \cup \{1, 2\}\)[/tex] means combining all elements from set [tex]\(X\)[/tex] and the set [tex]\(\{1, 2\}\)[/tex] without repeating any element.
2. Identify the Full Set: We want the result of this union to be [tex]\(\{1, 2, 3, 5, 7, 9\}\)[/tex].
3. Determine Missing Elements: Start with the set [tex]\(\{1, 2, 3, 5, 7, 9\}\)[/tex] and remove the elements \{1, 2\}, because these are already part of both sets in the union:
- Remove 1 and 2 from \{1, 2, 3, 5, 7, 9\}.
4. Find Set [tex]\(X\)[/tex]: After removing \{1, 2\}, we are left with \{3, 5, 7, 9\}. These are the elements that must be in set [tex]\(X\)[/tex] to complete the union with \{1, 2\} and reach \{1, 2, 3, 5, 7, 9\}.
5. Identify the Smallest Set: Therefore, the smallest set [tex]\(X\)[/tex] that satisfies the equation is \{3, 5, 7, 9\}.
By following these steps, we determine that the correct choice is option (d) [tex]\(\{3, 5, 7, 9\}\)[/tex].
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