Given the number cube with faces numbered 1 to 6, let's analyze the options one by one:
1. If [tex]\( A \)[/tex] is a subset of [tex]\( S \)[/tex], [tex]\( A \)[/tex] could be [tex]\(\{0,1,2\}\)[/tex]:
Incorrect. The set [tex]\( S \)[/tex] contains only the numbers 1 to 6, so a subset [tex]\( A \)[/tex] cannot include numbers outside this range. The number 0 is not in [tex]\( S \)[/tex].
2. If [tex]\( A \)[/tex] is a subset of [tex]\( S \)[/tex], [tex]\( A \)[/tex] could be [tex]\(\{5,6\}\)[/tex]:
Correct. The set [tex]\( S \)[/tex] is [tex]\(\{1,2,3,4,5,6\}\)[/tex], and [tex]\(\{5,6\}\)[/tex] is a valid subset of this set.
3. If a subset [tex]\( A \)[/tex] represents the complement of rolling a 5, then [tex]\( A = \{1,2,3,4,6\}\)[/tex]:
Correct. The complement of rolling a 5 includes all the outcomes except for 5. Therefore, [tex]\( A \)[/tex] would be [tex]\(\{1,2,3,4,6\}\)[/tex].
4. If a subset [tex]\( A \)[/tex] represents the complement of rolling an even number, then [tex]\( A = \{1,3\}\)[/tex]:
Incorrect. The set of even numbers on the cube is [tex]\(\{2, 4, 6\}\)[/tex]. The complement would be the set of all other numbers, which are the odd numbers. Therefore, [tex]\( A \)[/tex] should be [tex]\(\{1, 3, 5\}\)[/tex].
So, the correct options 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].
