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Questions and Answers
Which of the following correctly describes a singleton set?
Which of the following correctly describes a singleton set?
- A set that contains an infinite number of elements.
- A set that contains more than one element.
- A set that contains exactly one element. (correct)
- A set that contains no elements.
What is the cardinal number of an empty set?
What is the cardinal number of an empty set?
- 0 (correct)
- 1
- Undefined
- Infinite
If set A is a subset of set B, which of the following statements is true?
If set A is a subset of set B, which of the following statements is true?
- A and B must have the same number of elements
- A must contain exactly half of the elements of B
- A cannot contain any elements
- All elements of A are in B, but B can have additional elements (correct)
How many elements are in the power set of a set with 3 elements?
How many elements are in the power set of a set with 3 elements?
What distinguishes a proper subset from a regular subset?
What distinguishes a proper subset from a regular subset?
Which of the following best describes an infinite set?
Which of the following best describes an infinite set?
What can be concluded about two sets A and B that are equivalent?
What can be concluded about two sets A and B that are equivalent?
In set theory, which is the definition of a universal set?
In set theory, which is the definition of a universal set?
Which operation represents the elements that are in either set A or set B, but not in both?
Which operation represents the elements that are in either set A or set B, but not in both?
If sets A and B are disjoint, what is the result of A ∩ B?
If sets A and B are disjoint, what is the result of A ∩ B?
What does the complement of set A consist of?
What does the complement of set A consist of?
Which of the following laws states that the union of a set with itself results in the same set?
Which of the following laws states that the union of a set with itself results in the same set?
What is the formula for finding the number of elements in the symmetric difference of sets A and B?
What is the formula for finding the number of elements in the symmetric difference of sets A and B?
If n(A) = 6 and n(B) = 5, and their intersection n(A ∩ B) = 3, what is n(A ∪ B)?
If n(A) = 6 and n(B) = 5, and their intersection n(A ∩ B) = 3, what is n(A ∪ B)?
Which of the following best describes the difference of two sets, A - B?
Which of the following best describes the difference of two sets, A - B?
According to De-Morgan's law, what is the complement of the union of two sets?
According to De-Morgan's law, what is the complement of the union of two sets?
What is the number of students who speak Hindi only?
What is the number of students who speak Hindi only?
How many elements are in the union of sets A and B if n(A) = 250, n(B) = 300, and n(A ∩ B) = 350?
How many elements are in the union of sets A and B if n(A) = 250, n(B) = 300, and n(A ∩ B) = 350?
What is the complement of the set (A – B) where A = {1, 2, 3, 4} and B = {2, 4, 6, 8}?
What is the complement of the set (A – B) where A = {1, 2, 3, 4} and B = {2, 4, 6, 8}?
How many elements does the set A contain if A consists of all ordered pairs (x, y) such that x is a factor of y and x ∈ {2, 3, 4}, y ∈ {4, 6, 9, 10}?
How many elements does the set A contain if A consists of all ordered pairs (x, y) such that x is a factor of y and x ∈ {2, 3, 4}, y ∈ {4, 6, 9, 10}?
Which of the following is a null set?
Which of the following is a null set?
How many teachers teach only Mathematics if there are 30 teachers in total, with 20 teaching Mathematics, 15 teaching Physics, and 5 teaching both subjects?
How many teachers teach only Mathematics if there are 30 teachers in total, with 20 teaching Mathematics, 15 teaching Physics, and 5 teaching both subjects?
If A = {x: x is an odd integer} and B = {x: x² - 8x + 15 = 0}, what is the relationship between A and B?
If A = {x: x is an odd integer} and B = {x: x² - 8x + 15 = 0}, what is the relationship between A and B?
Which of the following characteristics is not true about the set of natural numbers?
Which of the following characteristics is not true about the set of natural numbers?
In a class of 110 students, if $x$ students take both mathematics and $2x + 20$ students take statistics, what is the equation that represents the total number of students taking at least one of the subjects?
In a class of 110 students, if $x$ students take both mathematics and $2x + 20$ students take statistics, what is the equation that represents the total number of students taking at least one of the subjects?
If 105 students are taking examinations in English and mathematics, and 10 students fail in both subjects, how many students pass both subjects?
If 105 students are taking examinations in English and mathematics, and 10 students fail in both subjects, how many students pass both subjects?
What is the maximum number of students that could possibly pass either English or mathematics in the examination?
What is the maximum number of students that could possibly pass either English or mathematics in the examination?
In the set defined as $A = {x : x is a real number, x > 1 }$, which of the following statements is true?
In the set defined as $A = {x : x is a real number, x > 1 }$, which of the following statements is true?
With reference to sets, what would the intersection of $A$ and $B$, where $A = {2, 4, 6, 8, 10}$ and $B = {5, 10, 15, 20, 25}$, yield?
With reference to sets, what would the intersection of $A$ and $B$, where $A = {2, 4, 6, 8, 10}$ and $B = {5, 10, 15, 20, 25}$, yield?
How many students fail only in mathematics if 75 students pass in mathematics and 80 pass in English, given that 10 fail both?
How many students fail only in mathematics if 75 students pass in mathematics and 80 pass in English, given that 10 fail both?
What is x if $x$ represents the number of students taking both mathematics, and $2x + 20$ represents students taking statistics, given that the total is 110?
What is x if $x$ represents the number of students taking both mathematics, and $2x + 20$ represents students taking statistics, given that the total is 110?
In the context of students passing subjects, if 80 students pass English and 75 pass mathematics with 10 failing both, how many students pass only one subject?
In the context of students passing subjects, if 80 students pass English and 75 pass mathematics with 10 failing both, how many students pass only one subject?
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Study Notes
Set Theory Basics
- A singleton set contains only one element.
- The cardinal number of an empty set (∅) is 0.
- If set A is a subset of set B, then all elements of set A are also present in set B.
- The power set of a set with 3 elements has 2^3 = 8 elements.
- A proper subset is a subset that does not contain all the elements of the original set.
- An infinite set contains an unlimited number of elements that cannot be counted.
- Two equivalent sets have the same number of elements, even if the elements themselves are different.
- A universal set contains all possible elements relevant to a particular context.
- The union of sets A and B (A ∪ B) represents all elements present in either set A or set B.
- The intersection of disjoint sets A and B (A ∩ B) results in an empty set.
- The complement of set A (A') consists of all elements that are not in A but are present in the universal set.
- The idempotent law states that the union of a set with itself results in the same set (A ∪ A = A).
- The formula for the number of elements in the symmetric difference of sets A and B is: n(A Δ B) = n(A) + n(B) - 2n(A ∩ B).
- Given n(A) = 6, n(B) = 5, and n(A ∩ B) = 3, n(A ∪ B) is calculated as 6 + 5 - 3 = 8.
- The difference of two sets, A - B, consists of elements present in set A but not in set B.
- De-Morgan's law states that the complement of the union of two sets is equal to the intersection of their complements: (A ∪ B)' = A' ∩ B'.
Set Theory Problems
- Given 30 teachers, 20 teach mathematics, 15 teach physics, and 5 teach both, the number of teachers teaching only mathematics is 15.
- If A = {x: x is an odd integer} and B = {x: x² - 8x + 15 = 0}, then set B is a subset of set A.
- The set of natural numbers does not include negative numbers or zero.
- In a class of 110 students, if $x$ students take both mathematics and $2x + 20$ students take statistics, the equation representing the total number of students taking at least one subject is: x + (2x + 20) - x = 110.
- If 105 students are taking examinations in English and mathematics, and 10 students fail in both subjects, 95 students pass both subjects.
- The maximum number of students that could possibly pass either English or mathematics is 105.
- In the set defined as $A = {x : x is a real number, x > 1 }$, all elements are greater than 1.
- The intersection of $A = {2, 4, 6, 8, 10}$ and $B = {5, 10, 15, 20, 25}$ is {10}.
- Given 75 students pass mathematics, 80 pass English, and 10 fail both, 15 students fail only in mathematics.
- In a class of 110 students with $x$ students taking both mathematics and $2x + 20$ students taking statistics, x = 30.
- If 80 students pass English, 75 pass mathematics, and 10 fail both, 145 students pass only one subject.
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