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]
If [tex]A[/tex] is a subset of [tex]S[/tex], [tex]A[/tex] could be [tex]{0,1,2}[/tex].
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].
If a subset [tex]A[/tex] represents the complement of rolling an even number, then [tex]A={1,3}[/tex].
Answer (1)
A subset can only contain elements from the original set.
The complement of an event includes all outcomes not in the event.
The statement 'If A is a subset of S , A could be { 5 , 6 } ' is true.
The statement 'If a subset A represents the complement of rolling a 5 , then A = { 1 , 2 , 3 , 4 , 6 } ' is true.
The statement 'If a subset A represents the complement of rolling an even number, then A = { 1 , 3 , 5 } ' is true.
The true statements are: { 5 , 6 } , { 1 , 2 , 3 , 4 , 6 } , and { 1 , 3 , 5 } .
The question asks to select three options. The correct options 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 } '. The third correct option is 'If a subset A represents the complement of rolling an even number, then A = { 1 , 3 , 5 } '. However, the last option in the question is 'If a subset A represents the complement of rolling an even number, then A = { 1 , 3 } ', which is not true. There seems to be an error in the question. Assuming that the last option should be 'If a subset A represents the complement of rolling an even number, then A = { 1 , 3 , 5 } ', the answer would be { 5 , 6 } , { 1 , 2 , 3 , 4 , 6 } , and { 1 , 3 , 5 } . Since the last option is not correct, we can only select two options. The question asks to select three options, so there must be an error in the question. The correct options based on the given choices are: { 5 , 6 } and { 1 , 2 , 3 , 4 , 6 } .
Explanation
Analyze the problem We are given a sample space S = { 1 , 2 , 3 , 4 , 5 , 6 } representing the possible outcomes of rolling a number cube. We need to determine which of the given statements about subsets and complements are true.
Evaluate the first statement Let's evaluate the first statement: '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 . Therefore, this statement is false.
Evaluate the second statement Now, let's evaluate the second statement: 'If A is a subset of S , A could be { 5 , 6 } '. Both 5 and 6 are elements of S , so the set { 5 , 6 } is a valid subset of S . Therefore, this statement is true.
Evaluate the third statement Next, let's evaluate the third statement: 'If a subset A represents the complement of rolling a 5, then A = { 1 , 2 , 3 , 4 , 6 } '. The complement of rolling a 5 includes all elements in S that are not equal to 5. This is the set { 1 , 2 , 3 , 4 , 6 } . Therefore, this statement is true.
Evaluate the fourth statement Finally, let's evaluate the fourth statement: 'If a subset A represents the complement of rolling an even number, then A = { 1 , 3 } '. The even numbers in S are 2 , 4 , and 6 . The complement of rolling an even number is the set of all elements in S that are not even, which is { 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 and statistics. For instance, if you're analyzing the outcomes of a game, you might want to know the probability of not rolling a specific number. This involves identifying the complement of that event, which is the set of all other possible outcomes. Knowing how to correctly identify subsets and complements allows you to accurately calculate probabilities and make informed decisions based on data.
