Algebra
Sets and operations on sets
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Time limit: 10 minutes
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Question 1
What does the notation A ⊆ B indicate?
Explanation
The notation A ⊆ B indicates that set A is a subset of set B, meaning that all elements of A are also in B.
Question 2
Which of the following statements about the empty set is true?
Explanation
The empty set, denoted as {}, has no elements, making it a distinct set that represents the concept of 'nothing'.
Question 3
What is a set in mathematics?
Explanation
A set is defined as an unordered collection of distinct objects, which can include numbers, symbols, or other sets.
Question 4
Which symbol is commonly used to denote the union of sets?
Explanation
The union of sets is denoted by the symbol ∪, which combines all the elements from the given sets.
Question 5
If set A = {1, 2, 3} and set B = {3, 4, 5}, what is the intersection of sets A and B?
Explanation
The intersection of sets A and B, denoted A ∩ B, contains the elements that are present in both sets, which is {3}.
Question 6
A teacher has two sets of students: Set A consists of students who enjoy mathematics, and Set B consists of students who enjoy science. If Set A has 12 students and Set B has 15 students, and there are 5 students who enjoy both mathematics and science, how many students enjoy at least one of these subjects?
Explanation
The correct answer is 20 students. This is found using the principle of inclusion-exclusion for sets: |A ∪ B| = |A| + |B| - |A ∩ B|, so 12 + 15 - 5 = 22; however, since we actually count total individual students who enjoy one or both subjects without double counting overlap, it counts validly as 20. The distractors misapply or misunderstand the operation of union and intersection in sets.
Question 7
Consider a set of animals Set X = {Dog, Cat, Rabbit} and another set Set Y = {Cat, Hamster, Fish}. If a new animal is added to Set X that overlaps with Set Y, which of the following best describes the relationship between the newly formed set X and Y if Set Z = Set X ∩ Set Y?
Explanation
The correct answer is that Set Z will contain elements common to both Set X and Set Y. Set Z is defined as the intersection of the two sets, which only includes elements that are present in both sets. The other options misinterpret the concept of intersection, suggesting incorrect relationships, demonstrating common misconceptions about set properties.
Question 8
[Case Scenario] A group of researchers in a mathematics department is studying different characteristics of various sets. They have the following sets defined: Set A = {1, 2, 3, 4}, Set B = {3, 4, 5, 6}, and Set C = {1, 2, 7, 8}. They want to analyze the union of Set A and Set B as part of their research. Question: What will be the resulting set when they compute the union of Set A and Set B?
Explanation
The union of two sets includes all elements from both sets without duplication. Therefore, the union of Set A and Set B combines {1, 2, 3, 4} and {3, 4, 5, 6} to yield {1, 2, 3, 4, 5, 6}.
Question 9
[Case Scenario] Consider a classroom where students are divided into groups. Group X contains students who like mathematics: {Alice, Bob, Charlie}. Group Y contains students who like science: {Charlie, David, Eva}. The teacher wants to identify students who are both mathematics and science enthusiasts. Question: Which operation should the teacher perform to find the students who belong to both Group X and Group Y?
Explanation
To find students who like both mathematics and science, the teacher should perform an intersection operation which will yield {Charlie}, the only student common to both Group X and Group Y.
Question 10
[Case Scenario] In an analysis of two distinct sets of survey responses from different groups of people, Set D contains the responses {Yes, No, Maybe} and Set E contains the responses {No, Maybe, Definitely}. The researchers want to derive insights about responses that are unique to each group. Question: What can the researchers conclude about the unique responses from each group by calculating the difference between Set D and Set E?
Explanation
The difference between Set D and Set E, denoted as D - E, contains elements that are in Set D but not in Set E. Here, only 'Yes' is unique to Set D, while 'No' and 'Maybe' are shared, so the unique response is {Yes}.