A number cube has faces numbered 1 to 6. What is true about rolling the number cube one time? Select three options.
[tex]S={1,2,3,4,5,6}[/tex]
A. If [tex]A[/tex] is a subset of [tex]S[/tex], [tex]A[/tex] could be [tex]{0,1,2}[/tex].
B. If [tex]A[/tex] is a subset of [tex]S[/tex], [tex]A[/tex] could be [tex]{5,6}[/tex].
C. If a subset [tex]A[/tex] represents the complement of rolling a 5, then [tex]A={1,2,3,4,6}[/tex].
D. If a subset [tex]A[/tex] represents the complement of rolling an even number, then [tex]A={1,3}[/tex].
Answer (1)
A subset of a set can only contain elements from the original set.
The complement of an event includes all outcomes in the sample space that are not in the event.
The subset { 5 , 6 } is a valid subset of S .
The complement of rolling a 5 is { 1 , 2 , 3 , 4 , 6 } .
Explanation
Analyze the problem and sample space We are given a number cube with faces numbered 1 to 6. This means our sample space, denoted by S , is the set of possible outcomes when rolling the cube once: S = { 1 , 2 , 3 , 4 , 5 , 6 } . We need to determine which of the given statements about subsets of S and complements of events are true.
Evaluate the first statement The first statement says: If A is a subset of S , A could be { 0 , 1 , 2 } . A subset of S can only contain elements that are also in S . Since 0 is not an element of S , the set { 0 , 1 , 2 } cannot be a subset of S . Thus, this statement is false.
Evaluate the second statement The second statement says: If A is a subset of S , A could be { 5 , 6 } . Since both 5 and 6 are elements of S , the set { 5 , 6 } can be a subset of S . Thus, this statement is true.
Evaluate the third statement The third statement says: If a subset A represents the complement of rolling a 5, then A = { 1 , 2 , 3 , 4 , 6 } . The complement of rolling a 5 is the set of all outcomes in S that are not 5. This is indeed the set { 1 , 2 , 3 , 4 , 6 } . Thus, this statement is true.
Evaluate the fourth statement The fourth statement says: If a subset A represents the complement of rolling an even number, then A = { 1 , 3 } . The even numbers in S are { 2 , 4 , 6 } . The complement of rolling an even number is the set of all outcomes in S that are not even. This is the set { 1 , 3 , 5 } . Since the given set is { 1 , 3 } , this statement is false.
Identify the true statements Therefore, the true statements are:
If A is a subset of S , A could be { 5 , 6 } .
If a subset A represents the complement of rolling a 5, then A = { 1 , 2 , 3 , 4 , 6 } .
Examples
Understanding subsets and complements is crucial in probability. For instance, if you're analyzing the odds of winning a lottery, you need to identify the set of all possible outcomes (the sample space) and the specific outcomes that lead to a win (a subset). The complement would then be the set of outcomes that lead to a loss. This helps in calculating the probability of winning versus losing.
