Let’s analyze each of the given statements one by one in detail to determine their truthfulness with respect to the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
1. Statement: [tex]\( 4 \in B \)[/tex]
- Set [tex]\( B \)[/tex] is given as [tex]\( B = \{3, 2, 1\} \)[/tex].
- We need to check if the element [tex]\( 4 \)[/tex] is present in set [tex]\( B \)[/tex].
- By examining the elements of set [tex]\( B \)[/tex], we see that [tex]\( 4 \)[/tex] is not included in set [tex]\( B \)[/tex].
- Therefore, the statement [tex]\( 4 \in B \)[/tex] is False.
2. Statement: [tex]\( A \subseteq B \)[/tex]
- Set [tex]\( A \)[/tex] is [tex]\( \{1, 2, 3\} \)[/tex] and set [tex]\( B \)[/tex] is [tex]\( \{3, 2, 1\} \)[/tex].
- We need to determine if every element of set [tex]\( A \)[/tex] is also an element of set [tex]\( B \)[/tex].
- Here, [tex]\( A \)[/tex] contains the elements [tex]\( 1, 2, \)[/tex] and [tex]\( 3 \)[/tex], and set [tex]\( B \)[/tex] also contains [tex]\( 1, 2, \)[/tex] and [tex]\( 3 \)[/tex].
- Since all elements of [tex]\( A \)[/tex] are in [tex]\( B \)[/tex], the statement [tex]\( A \subseteq B \)[/tex] is True.
3. Statement: [tex]\( A \)[/tex] is an infinite set
- Set [tex]\( A \)[/tex] is explicitly given as [tex]\( \{1, 2, 3\} \)[/tex], which contains three elements.
- Since set [tex]\( A \)[/tex] has a finite number of elements, it is a finite set, not an infinite set.
- Therefore, the statement [tex]\( A \)[/tex] is an infinite set is False.
4. Statement: none of the above
- This statement implies that all other statements would need to be false to be true itself.
- Given that statement [tex]\( A \subseteq B \)[/tex] is true, not all statements are false.
Summarizing our analysis:
- [tex]\( 4 \in B \)[/tex] is False.
- [tex]\( A \subseteq B \)[/tex] is True.
- [tex]\( A \)[/tex] is an infinite set is False.
- "none of the above" is also False because [tex]\( A \subseteq B \)[/tex] is True.
Given these points, the correct answer to which statement is true is:
[tex]\[ \boxed{A \subseteq B} \][/tex]
