Let's consider each of the statements one by one to determine if they are true.
1. Statement: "If [tex]\( A \)[/tex] is a subset of [tex]\( S \)[/tex], [tex]\( A \)[/tex] could be [tex]\( \{0, 1, 2\} \)[/tex]."
Explanation: The set [tex]\( S \)[/tex] contains the numbers [tex]\( \{1, 2, 3, 4, 5, 6\} \)[/tex]. A subset of [tex]\( S \)[/tex] means that all elements of [tex]\( A \)[/tex] must be in [tex]\( S \)[/tex]. Since 0 is not in [tex]\( S \)[/tex], [tex]\( \{0, 1, 2\} \)[/tex] cannot be a subset of [tex]\( S \)[/tex].
Conclusion: This statement is false.
2. Statement: "If [tex]\( A \)[/tex] is a subset of [tex]\( S \)[/tex], [tex]\( A \)[/tex] could be [tex]\( \{5, 6\} \)[/tex]."
Explanation: The set [tex]\( S \)[/tex] contains the numbers [tex]\( \{1, 2, 3, 4, 5, 6\} \)[/tex]. The numbers 5 and 6 are both in [tex]\( S \)[/tex], so [tex]\( \{5, 6\} \)[/tex] is indeed a subset of [tex]\( S \)[/tex].
Conclusion: 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]."
Explanation: The complement of rolling a 5 means all outcomes except for 5. The set [tex]\( S \)[/tex] is [tex]\( \{1, 2, 3, 4, 5, 6\} \)[/tex], so excluding 5, we get [tex]\( \{1, 2, 3, 4, 6\} \)[/tex].
Conclusion: 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]."
Explanation: The even numbers in [tex]\( S \)[/tex] are 2, 4, and 6. The complement of rolling an even number means all outcomes except for 2, 4, and 6. Thus, the complement of rolling an even number is [tex]\( \{1, 3, 5\} \)[/tex].
Conclusion: This statement is false.
So, the three true statements 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]."
