Let's analyze each statement one-by-one to determine whether it is true or not.
1. Statement: If [tex]\(A \)[/tex] is a subset of [tex]\(S, A \)[/tex] could be [tex]\(\{0, 1, 2\}\)[/tex].
To be a subset of [tex]\(S = \{1, 2, 3, 4, 5, 6\}\)[/tex], every element in set [tex]\(A\)[/tex] must also be an element of set [tex]\(S\)[/tex]. The set [tex]\(\{0, 1, 2\}\)[/tex] includes the element 0, which is not in [tex]\(S\)[/tex].
Therefore, this statement is false.
2. Statement: If [tex]\(A\)[/tex] is a subset of [tex]\(S, A\)[/tex] could be [tex]\(\{5, 6\}\)[/tex].
To be a subset of [tex]\(S\)[/tex], every element in the set [tex]\(A\)[/tex] must also be an element of [tex]\(S\)[/tex]. The set [tex]\(\{5, 6\}\)[/tex] includes elements 5 and 6, both of which are in [tex]\(S\)[/tex].
Therefore, this statement is true.
3. Statement: If a subset [tex]\(A\)[/tex] represents the complement of rolling a 5, then [tex]\(A = \{1, 2, 3, 4, 6\}\)[/tex].
The complement of rolling a 5 means every possible outcome except rolling a 5. The set of all outcomes except 5 is [tex]\(\{1, 2, 3, 4, 6\}\)[/tex].
Therefore, this statement is true.
4. Statement: If a subset [tex]\(A\)[/tex] represents the complement of rolling an even number, then [tex]\(A = \{1, 3\}\)[/tex].
The even numbers in the set [tex]\(S = \{1, 2, 3, 4, 5, 6\}\)[/tex] are \{2, 4, 6\}. The complement of rolling an even number should include all outcomes that are not even numbers, thus it consists of the odd numbers. The odd numbers in the set are \{1, 3, 5\}.
Therefore, this statement is false.
To conclude, the statements that are true are:
- If [tex]\(A\)[/tex] is a subset of [tex]\(S\)[/tex], [tex]\(A\)[/tex] could be [tex]\(\{5, 6\}\)[/tex].
- If a subset [tex]\(A\)[/tex] represents the complement of rolling a 5, then [tex]\(A = \{1, 2, 3, 4, 6\}\)[/tex].
These are the steps and conclusions based on the analysis of each statement.
