A spinner is divided into eight equal-sized sections, numbered from 1 to 8, inclusive. Which statements are true about spinning the spinner one time? Choose three correct answers.
* If a subset [tex]$A$[/tex] represents spinning a number less than 4, then [tex]$A={1,2,3,4}$[/tex].
* If A is a subset of [tex]$S$[/tex], A could be [tex]${1,2,3}$[/tex].
* If A is a subset of [tex]$S$[/tex], A could be [tex]${7,8,9}$[/tex].
* [tex]$S={1,2,3,4,5,6,7,8}$[/tex]
* If a subset A represents the complement of spinning an odd number, then [tex]$A ={2,4,6,8}$[/tex].
Answer (2)
Statement 2 is correct because { 1 , 2 , 3 } is a subset of S = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } .
Statement 4 is correct because the complement of odd numbers in S is { 2 , 4 , 6 , 8 } .
Statements 1 and 3 are incorrect.
The problem asks for three correct answers, but only two statements are correct, indicating a possible error in the question.
Explanation
Analyze the problem We are given a spinner divided into eight equal sections, numbered from 1 to 8. We need to determine which of the given statements about subsets of the possible outcomes are correct. Let's analyze each statement.
Evaluate Statement 1 Statement 1: If a subset A represents spinning a number less than 4, then A = { 1 , 2 , 3 , 4 } .
Numbers less than 4 are 1, 2, and 3. Therefore, the correct subset should be A = { 1 , 2 , 3 } . The given statement is incorrect because it includes 4, which is not less than 4.
Evaluate Statement 2 Statement 2: If A is a subset of S , A could be { 1 , 2 , 3 } .
Since S = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } , and all elements in { 1 , 2 , 3 } are also in S , this statement is correct. A subset can contain any combination of elements from the original set, including all or none of them.
Evaluate Statement 3 Statement 3: If A is a subset of S , A could be { 7 , 8 , 9 } .
Since S = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } , and the element 9 is not in S , this statement is incorrect. A subset can only contain elements that are present in the original set.
Evaluate Statement 4 Statement 4: If a subset A represents the complement of spinning an odd number, then A = { 2 , 4 , 6 , 8 } .
The odd numbers in S are { 1 , 3 , 5 , 7 } . The complement of these numbers within S is all the numbers in S that are not odd, which are { 2 , 4 , 6 , 8 } . Therefore, this statement is correct.
Identify Correct Statements The correct statements are Statement 2 and Statement 4. We need to choose three correct answers, but only two statements are correct. Let's re-evaluate Statement 1. The statement says "less than 4", which means strictly less than 4. So, A = {1, 2, 3}. The given set is A = {1, 2, 3, 4}, which is incorrect. Statement 3 says A could be {7, 8, 9}. Since 9 is not in S, this is incorrect. Statement 4 says A represents the complement of spinning an odd number, so A = {2, 4, 6, 8}, which is correct. Therefore, the correct statements are 2 and 4. Since we need to choose three correct answers, there must be an error in the problem statement. Assuming the problem meant to ask for all correct statements, the correct statements are 2 and 4. However, since we must choose three, and only two are correct, there seems to be an issue with the question itself.
Final Answer Since the problem asks to choose three correct answers and only two statements are correct (Statement 2 and Statement 4), there must be an error in the problem. However, based on our analysis, the correct statements are:
If A is a subset of S, A could be {1, 2, 3}.
If a subset A represents the complement of spinning an odd number, then A = {2, 4, 6, 8}.
Examples
Understanding subsets and complements is crucial in probability and statistics. For example, if you're analyzing the results of a survey, you might define a subset as the group of respondents who answered 'yes' to a particular question. The complement would then be the group who answered 'no' or didn't answer the question. Analyzing these subsets helps you draw meaningful conclusions from the data.
The correct statements regarding spinning the spinner are Statements 2, 4, and 5. Statement 1 is incorrect because it includes 4, and Statement 3 is incorrect because 9 is not part of the set. There appears to be an error in the question since only three statements can be valid.
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