Set Theory Overview
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Set Theory Overview

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Questions and Answers

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?

  • 0 (correct)
  • 1
  • Undefined
  • Infinite

    • 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?

    <p>8</p> Signup and view all the answers

    What distinguishes a proper subset from a regular subset?

    <p>A proper subset cannot be the same as the original set.</p> Signup and view all the answers

    Which of the following best describes an infinite set?

    <p>A set that cannot be counted due to having unlimited elements.</p> Signup and view all the answers

    What can be concluded about two sets A and B that are equivalent?

    <p>They can have different elements, but the same number of elements.</p> Signup and view all the answers

    In set theory, which is the definition of a universal set?

    <p>A set that contains all objects including itself.</p> Signup and view all the answers

    Which operation represents the elements that are in either set A or set B, but not in both?

    <p>A ∆ B</p> Signup and view all the answers

    If sets A and B are disjoint, what is the result of A ∩ B?

    <p>ϕ</p> Signup and view all the answers

    What does the complement of set A consist of?

    <p>Elements in U that are not in A</p> Signup and view all the answers

    Which of the following laws states that the union of a set with itself results in the same set?

    <p>Idempotent law</p> Signup and view all the answers

    What is the formula for finding the number of elements in the symmetric difference of sets A and B?

    <p>n(A ∆ B) = n(A) + n(B) - 2n(A ∩ B)</p> Signup and view all the answers

    If n(A) = 6 and n(B) = 5, and their intersection n(A ∩ B) = 3, what is n(A ∪ B)?

    <p>8</p> Signup and view all the answers

    Which of the following best describes the difference of two sets, A - B?

    <p>Elements in A that are not in B</p> Signup and view all the answers

    According to De-Morgan's law, what is the complement of the union of two sets?

    <p>U - (A ∪ B)</p> Signup and view all the answers

    What is the number of students who speak Hindi only?

    <p>300</p> Signup and view all the answers

    How many elements are in the union of sets A and B if n(A) = 250, n(B) = 300, and n(A ∩ B) = 350?

    <p>200</p> Signup and view all the answers

    What is the complement of the set (A – B) where A = {1, 2, 3, 4} and B = {2, 4, 6, 8}?

    <p>{2, 4, 5, 6, 7, 8}</p> Signup and view all the answers

    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}?

    <p>10</p> Signup and view all the answers

    Which of the following is a null set?

    <p>C = {ϕ}</p> Signup and view all the answers

    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?

    <p>10</p> Signup and view all the answers

    If A = {x: x is an odd integer} and B = {x: x² - 8x + 15 = 0}, what is the relationship between A and B?

    <p>B ⊆ A</p> Signup and view all the answers

    Which of the following characteristics is not true about the set of natural numbers?

    <p>Closed under subtraction</p> Signup and view all the answers

    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?

    <p>$x + (2x + 20)$</p> Signup and view all the answers

    If 105 students are taking examinations in English and mathematics, and 10 students fail in both subjects, how many students pass both subjects?

    <p>65</p> Signup and view all the answers

    What is the maximum number of students that could possibly pass either English or mathematics in the examination?

    <p>95</p> Signup and view all the answers

    In the set defined as $A = {x : x is a real number, x > 1 }$, which of the following statements is true?

    <p>The set is empty.</p> Signup and view all the answers

    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?

    <p>${10}$</p> Signup and view all the answers

    How many students fail only in mathematics if 75 students pass in mathematics and 80 pass in English, given that 10 fail both?

    <p>25</p> Signup and view all the answers

    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?

    <p>25</p> Signup and view all the answers

    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?

    <p>26</p> Signup and view all the answers

    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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