Set
Set problems
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Time limit: 10 minutes
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All questions must be answered to submit
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Question 1
What does it mean to intersect two sets?
Explanation
The intersection of two sets consists of all elements that are common to both sets.
Question 2
What is a set problem?
Explanation
A set problem involves grouping elements based on specific criteria or relationships, often exploring how these elements interact.
Question 3
Which of the following is an example of a set?
Explanation
A set is typically defined as a collection of distinct objects, considered as an object in its own right, such as the collection of all whole numbers.
Question 4
Which symbol represents the union of two sets?
Explanation
The union of two sets is represented by the symbol '∪' which signifies the combination of elements from both sets without duplicates.
Question 5
If Set A contains {1, 2, 3} and Set B contains {3, 4, 5}, what is the intersection of Set A and Set B?
Explanation
The intersection of Set A and Set B is {3} as it is the only element common to both sets.
Question 6
A group of students is given a set problem to solve in class, where they need to allocate a limited budget to different projects. If they mistakenly allocate too much to one project based on incorrect assumptions about costs, what should they do next to rectify their decision based on set problem-solving strategies?
Explanation
The correct course of action is to reassess the assumptions made about the project costs and gather more accurate data. This step reflects critical analysis of the problem and is essential to resolve the discrepancies in resource allocation. Continuing with the current allocation or disregarding other projects is not a productive strategy as it ignores the need for accurate information. Asking for a replacement decision undermines the learning opportunity of solving the problem independently.
Question 7
During a class discussion on set problems, a student proposes using a random selection method to determine project priorities rather than a systematic evaluation of their impact. What is a potential flaw in this approach based on principles of effective problem-solving?
Explanation
The potential flaw in using random selection is that it ignores the specific criteria needed for evaluating project impact, which is crucial for making informed decisions. While it may seem faster, it undermines a systematic approach essential for problem-solving. Claiming that it encourages diversity or yields innovative outcomes does not address the fundamental lack of analysis in prioritizing projects effectively.
Question 8
[Case Scenario] You are a mathematics teacher who has been working with a group of students struggling with problem-solving involving set theory. After several lessons, you notice that some students are still confused about how to apply the union and intersection of sets in practical problems. One day, you present the following problem: "In a class of 30 students, 18 like mathematics and 12 like science. If 5 students like both subjects, how many students like either mathematics or science?" As the students delve into solving the problem, you overhear various approaches. Question: What is the correct method to determine the number of students who like either mathematics or science?
Explanation
To find the number of students who like either subject, apply the principle of inclusion-exclusion. The formula for the union of two sets is |A ∪ B| = |A| + |B| - |A ∩ B|. Here, |A| is students who like mathematics (18), |B| is those who like science (12), and |A ∩ B| is those who like both (5). Therefore, |A ∪ B| = 18 + 12 - 5 = 25.
Question 9
[Case Scenario] During a workshop on set theory, attendees were divided into two groups based on their preference for either classical music or jazz. The first group contained 25 people who preferred classical music, and the second group had 15 who preferred jazz. It was found that 10 participants enjoyed both genres. The instructor posed the question: "How many unique participants were involved in the workshop?" As participants began to calculate the total, there were differing opinions on how to arrive at the answer. Question: How can the total number of unique participants be accurately determined?
Explanation
To determine the number of unique participants, one needs to apply the inclusion-exclusion principle specifically. The total unique participants can be calculated as |Classical| + |Jazz| - |Both| = 25 + 15 - 10, resulting in 30 unique individuals in the workshop.
Question 10
[Case Scenario] In a local survey, researchers asked 50 people whether they preferred spending time outdoors or watching movies. The results showed that 27 people preferred outdoor activities, 22 preferred movies, and 12 indicated that they enjoyed both activities. The researchers now want to present these findings in a clear way to visualize the preferences across their sample population. They are considering how to best represent this information graphically to highlight the overlaps and individual preferences. Question: What is the most effective way to represent the survey results?
Explanation
The best way to visually represent the survey results from the study is through a Venn diagram. This diagram allows viewers to easily see the distinct groups of people who prefer outdoor activities, those who prefer movies, and those who enjoy both activities, enhancing comprehension of the data's relational aspects.