Set Theory - Introduction
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Questions and Answers

What is a characteristic of the empty set?

  • It is the largest set in set theory.
  • It contains a single element.
  • It is a subset of every set. (correct)
  • It has elements in common with every other set.

    • If sets A and B are disjoint, which statement is true?

  • Both sets have at least one element in common.
  • Neither set is a subset of the other. (correct)
  • One of the sets must be empty.
  • Both sets can be infinite.

    • Which of the following is the correct representation of the universal set?

  • A single element set.
  • A rectangle enclosing all other sets. (correct)
  • A circle representing all subsets.
  • An empty set denoted as ∅.

    • How many empty sets exist in the context of set theory?

    <p>Only one empty set.</p> Signup and view all the answers

    What is the relationship between rational numbers and complex numbers?

    <p>Rational numbers are a subset of complex numbers.</p> Signup and view all the answers

    In a Venn diagram, if set A is a subset of set B, how are they represented?

    <p>A is entirely within B.</p> Signup and view all the answers

    Which of the following correctly describes disjoint sets?

    <p>They do not share any elements.</p> Signup and view all the answers

    What set notation is used for an empty set?

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

    What symbol is used to denote membership in a set?

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

    Which of the following correctly specifies the set of positive even integers greater than 0?

    <p>{x | x is an even integer, x &gt; 0}</p> Signup and view all the answers

    Which of the following statements about sets is false?

    <p>A set can contain duplicate elements.</p> Signup and view all the answers

    Which set is correctly defined as containing odd positive integers less than 10?

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

    If E = {x | x^2 - 3x + 2 = 0}, what are the elements of set E?

    <p>{1, 2}</p> Signup and view all the answers

    Which notation is used to list members of a set explicitly?

    <p>A = {1, 3, 5, ...}</p> Signup and view all the answers

    What does the vertical line '|' signify in set notation?

    <p>such that</p> Signup and view all the answers

    Which of the following describes the sets F and G if F = {2, 1} and G = {1, 2, 2, 1}?

    <p>F and G are identical sets with different representations.</p> Signup and view all the answers

    What is the definition of two sets being disjoint?

    <p>They have no elements in common.</p> Signup and view all the answers

    If sets A and B are disjoint, which statement is true regarding the union of A and B?

    <p>A ∪ B = A + B</p> Signup and view all the answers

    Which property correctly describes the intersection of sets A and B?

    <p>A ∩ B is a subset of A and B.</p> Signup and view all the answers

    What can be concluded if A is a subset of B?

    <p>A ∩ B = A.</p> Signup and view all the answers

    For sets A and B, which theorem states that A ⊆ B, A ∩ B = A, and A ∪ B = B are equivalent?

    <p>Theorem 1.4</p> Signup and view all the answers

    If U is the disjoint union of M and F, which of the following is true?

    <p>All students at the university are in either M or F.</p> Signup and view all the answers

    Which of the following statements about sets A and B is correct?

    <p>For any element in A ∪ B, it must belong to A or B.</p> Signup and view all the answers

    What is the intersection of sets A and C where A = {1, 2, 3, 4} and C = {2, 3, 8, 9}?

    <p>{2, 3}</p> Signup and view all the answers

    What is the result of applying DeMorgan’s Law to the expression $(A ∪ B)^C$?

    <p>$A^C ∩ B^C$</p> Signup and view all the answers

    Which logical equivalence represents the negation of a disjunction?

    <p>¬(p ∨ q) = ¬p ∧ ¬q</p> Signup and view all the answers

    What defines a finite set?

    <p>It is either empty or contains a specific positive integer number of elements.</p> Signup and view all the answers

    Which of the following sets is countably infinite?

    <p>The set of even positive integers</p> Signup and view all the answers

    What is the dual of the equation $(U ∩ A) ∪ (B ∩ A) = A$?

    <p>$(∅ ∪ A) ∩ (B ∪ A) = A$</p> Signup and view all the answers

    How is the notation $n(S)$ used in set theory?

    <p>It denotes the number of elements in set S.</p> Signup and view all the answers

    Which of the following sets is uncountable?

    <p>The unit interval $[0, 1]$</p> Signup and view all the answers

    What is the notation used to denote the empty set?

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

    What is the conclusion of Lemma 1.6 regarding the union of two finite disjoint sets A and B?

    <p>n(A ∪ B) = n(A) + n(B).</p> Signup and view all the answers

    When applying the Inclusion-Exclusion Principle to finite sets A and B, what adjustment is made to avoid overcounting?

    <p>Subtract n(A ∩ B) from the sum of n(A) and n(B).</p> Signup and view all the answers

    According to Corollary 1.7, what is the correct way to find the number of elements in set A that are not in set B?

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

    If set A contains 25 elements and overlaps with set B containing 10 elements, what represents n(A extbackslash B)?

    <p>25 - 10 = 15</p> Signup and view all the answers

    How can the count of the complement of a finite subset A within a universal set U be determined?

    <p>n(A^C) = n(U) - n(A)</p> Signup and view all the answers

    What is a significant property of the sets A and B stated in Lemma 1.6?

    <p>Sets A and B must be disjoint.</p> Signup and view all the answers

    In the context of the Inclusion–Exclusion Principle, which statement is correct?

    <p>It reduces the count by the intersection of the sets.</p> Signup and view all the answers

    What is the relationship between the sets A, B, and their intersection in the Inclusion–Exclusion Principle?

    <p>n(A ∪ B) is equal to n(A) plus n(B) minus n(A ∩ B).</p> Signup and view all the answers

    What does the complement of a set A, denoted by AC, represent?

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

    Which of the following notations is used for the relative complement of a set B with respect to A?

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

    How is the symmetric difference of two sets A and B denoted?

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

    If A = {1, 2, 3, 4} and B = {3, 4, 5, 6, 7}, what is the result of A extbackslash B?

    <p>{1, 2}</p> Signup and view all the answers

    In the expression A ⊕ B = (A ∪ B) extbackslash(A ∩ B), what does A ∪ B represent?

    <p>All elements in either A or B</p> Signup and view all the answers

    What is the result of the symmetric difference A ⊕ C if A = {1, 2, 3, 4} and C = {2, 3, 8, 9}?

    <p>{1, 4, 8, 9}</p> Signup and view all the answers

    Which mathematical operation is used to represent the fundamental product of sets A1, A2, ..., An?

    <p>A1 ∩ A2 ∩ ... ∩ An</p> Signup and view all the answers

    Which of the following correctly represents the relative complement of A with respect to the universal set U?

    <p>U extbackslash A</p> Signup and view all the answers

    Study Notes

    Set Theory - Introduction

    • Set theory is a fundamental concept in mathematics.
    • Sets are well-defined collections of objects.
    • Elements or members of a set are the objects within it.
    • Capital letters (A, B, X, Y) denote sets, while lowercase letters (a, b, x, y) denote elements.
    • "∈" denotes "is an element of," and "∉" denotes "is not an element of."

    Specifying Sets

    • Sets can be defined by listing their elements within braces { }.
    • Sets can also be defined by describing the properties that characterize their elements, with a letter (e.g., x) representing a typical member and a vertical line "|" to read as "such that."

    Examples

    • A = {1, 3, 5, 7, 9}
    • B = {x | x is an even integer and x > 0}

    Subsets

    • If every element of set A is also an element of set B, then A is a subset of B (A ⊆ B), or B contains A (B ⊃ A).
    • Two sets are equal if they have the same elements, and each is a subset of the other.
    • If A is not a subset of B (A ⊈ B), then at least one member of A does not belong to B.

    Universal Set

    • All sets under consideration are usually assumed to belong to a fixed large set, known as the universal set (U).

    Empty Set (Null Set)

    • A set with no elements is called the empty set (Ø).
    • There is only one empty set.

    Theorem 1.2

    • For any set A, the empty set (Ø) is a subset of A (Ø ⊆ A), and A is a subset of the universal set (A ⊆ U).

    Disjoint Sets

    • Disjoint sets have no elements in common.
    • Sets are disjoint unless some of the elements are common.

    Venn Diagrams

    • Venn diagrams are pictorial representations of sets.
    • The universal set is represented by a rectangle, and other sets by areas within the rectangle.
    • If A ⊆ B, then the disk for set A is entirely within the disk for set B.
    • If A and B are disjoint, then the disks are separated.

    Set Operations - Union

    • The union of sets A and B (A ∪ B) contains all elements that belong to A or to B (or both).
    • The resulting set contains all the elements of both the sets.

    Set Operations - Intersection

    • The intersection of sets A and B (A ∩ B) contains all elements that belong to both A and B.
    • The intersection contains common elements between the sets.

    Set Operations - Disjoint Union

    • The disjoint union of sets A and B (A ∪ B, where A ∩ B = Ø) contains all elements in A or B, but not both.
    • Sets have no common elements; they do not intersect.

    Set Operations - Complement

    • The complement of a set A (AC) is comprised of all elements in the universal set (U) that do not belong to A.

    Set Operations - Relative Complement

    • The relative complement of a set B with respect to set A (A\B), is the set of elements that belong to A but not to B.

    Set Operations - Symmetric Difference

    • The symmetric difference of sets A and B (A ⊕ B) contains elements that belong to either A or B, but not both.
    • The elements are present in either one of the sets, but not in both.

    Finite Sets

    • A set is finite if it is empty or contains a specific positive integer number of elements.

    Counting Principle

    • The notation n(S) or |S| represents the number of elements in a set S.

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    Lecture (1) Set Theory PDF

    Description

    Explore the foundational concepts of set theory, including defining sets, subsets, and the notation used. Understand the significance of elements within a set and learn how to represent sets using braces and properties. This quiz will assess your grasp of these fundamental mathematical concepts.

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