To determine which of the following sets is a subset of the set [tex]\(\{1, 2, 3\}\)[/tex], we need to examine whether every element of each given set is also an element of [tex]\(\{1, 2, 3\}\)[/tex].
Consider the sets provided:
1. [tex]\(\{1, 2, 3, 4\}\)[/tex]
2. [tex]\(\{0\}\)[/tex]
3. [tex]\(\{0, 1\}\)[/tex]
4. [tex]\(\{1, 2, 3\}\)[/tex]
Let's analyze each set one by one:
1. [tex]\(\{1, 2, 3, 4\}\)[/tex]:
- This set contains the elements 1, 2, 3, and 4.
- The elements 1, 2, and 3 are in [tex]\(\{1, 2, 3\}\)[/tex], but 4 is not.
- Since not all elements of [tex]\(\{1, 2, 3, 4\}\)[/tex] are in [tex]\(\{1, 2, 3\}\)[/tex], this set is not a subset of [tex]\(\{1, 2, 3\}\)[/tex].
2. [tex]\(\{0\}\)[/tex]:
- This set contains only the element 0.
- The element 0 is not in [tex]\(\{1, 2, 3\}\)[/tex].
- Since not all elements of [tex]\(\{0\}\)[/tex] are in [tex]\(\{1, 2, 3\}\)[/tex], this set is not a subset of [tex]\(\{1, 2, 3\}\)[/tex].
3. [tex]\(\{0, 1\}\)[/tex]:
- This set contains the elements 0 and 1.
- The element 1 is in [tex]\(\{1, 2, 3\}\)[/tex], but 0 is not.
- Since not all elements of [tex]\(\{0, 1\}\)[/tex] are in [tex]\(\{1, 2, 3\}\)[/tex], this set is not a subset of [tex]\(\{1, 2, 3\}\)[/tex].
4. [tex]\(\{1, 2, 3\}\)[/tex]:
- This set contains the elements 1, 2, and 3.
- All these elements are in [tex]\(\{1, 2, 3\}\)[/tex].
- Since all elements of [tex]\(\{1, 2, 3\}\)[/tex] are in [tex]\(\{1, 2, 3\}\)[/tex], this set is a subset of [tex]\(\{1, 2, 3\}\)[/tex].
After analyzing each set, we conclude that the only set that is a subset of [tex]\(\{1, 2, 3\}\)[/tex] is [tex]\(\{1, 2, 3\}\)[/tex]. Thus, the subset of [tex]\(\{1, 2, 3\}\)[/tex] is:
[tex]\[
\{1, 2, 3\}
\][/tex]
