To determine which of the given sets are subsets of [tex]\(\{1, 3, 5, 7, 9\}\)[/tex], we need to verify if every element in each set is also an element of [tex]\(\{1, 3, 5, 7, 9\}\)[/tex].
Let's examine each set individually:
1. [tex]\(\varnothing\)[/tex] (the empty set):
- By definition, the empty set is a subset of any set, including [tex]\(\{1, 3, 5, 7, 9\}\)[/tex].
2. Integers between 1 and 9:
- This set, in set notation, is [tex]\(\{1, 2, 3, 4, 5, 6, 7, 8, 9\}\)[/tex].
- Elements such as 2, 4, 6, and 8 are not in [tex]\(\{1, 3, 5, 7, 9\}\)[/tex].
- Therefore, this set is not a subset of [tex]\(\{1, 3, 5, 7, 9\}\)[/tex].
3. [tex]\(\{1, 2, 3\}\)[/tex]:
- We check each element: 1 and 3 are in [tex]\(\{1, 3, 5, 7, 9\}\)[/tex], but 2 is not.
- Therefore, [tex]\(\{1, 2, 3\}\)[/tex] is not a subset of [tex]\(\{1, 3, 5, 7, 9\}\)[/tex].
4. [tex]\(\{5, 6, 7\}\)[/tex]:
- We check each element: 5 and 7 are in [tex]\(\{1, 3, 5, 7, 9\}\)[/tex], but 6 is not.
- Therefore, [tex]\(\{5, 6, 7\}\)[/tex] is not a subset of [tex]\(\{1, 3, 5, 7, 9\}\)[/tex].
Based on the analysis:
- The empty set [tex]\(\varnothing\)[/tex] is a subset of [tex]\(\{1, 3, 5, 7, 9\}\)[/tex].
- The sets containing integers between 1 and 9, [tex]\(\{1, 2, 3\}\)[/tex], and [tex]\(\{5, 6, 7\}\)[/tex] are not subsets of [tex]\(\{1, 3, 5, 7, 9\}\)[/tex].
Therefore, the only set that is a subset of [tex]\(\{1, 3, 5, 7, 9\}\)[/tex] is the empty set [tex]\(\varnothing\)[/tex].
