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Questions and Answers
What is a characteristic of the empty set?
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?
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?
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?
How many empty sets exist in the context of set theory?
What is the relationship between rational numbers and complex numbers?
What is the relationship between rational numbers and complex numbers?
In a Venn diagram, if set A is a subset of set B, how are they represented?
In a Venn diagram, if set A is a subset of set B, how are they represented?
Which of the following correctly describes disjoint sets?
Which of the following correctly describes disjoint sets?
What set notation is used for an empty set?
What set notation is used for an empty set?
What symbol is used to denote membership in a set?
What symbol is used to denote membership in a set?
Which of the following correctly specifies the set of positive even integers greater than 0?
Which of the following correctly specifies the set of positive even integers greater than 0?
Which of the following statements about sets is false?
Which of the following statements about sets is false?
Which set is correctly defined as containing odd positive integers less than 10?
Which set is correctly defined as containing odd positive integers less than 10?
If E = {x | x^2 - 3x + 2 = 0}, what are the elements of set E?
If E = {x | x^2 - 3x + 2 = 0}, what are the elements of set E?
Which notation is used to list members of a set explicitly?
Which notation is used to list members of a set explicitly?
What does the vertical line '|' signify in set notation?
What does the vertical line '|' signify in set notation?
Which of the following describes the sets F and G if F = {2, 1} and G = {1, 2, 2, 1}?
Which of the following describes the sets F and G if F = {2, 1} and G = {1, 2, 2, 1}?
What is the definition of two sets being disjoint?
What is the definition of two sets being disjoint?
If sets A and B are disjoint, which statement is true regarding the union of A and B?
If sets A and B are disjoint, which statement is true regarding the union of A and B?
Which property correctly describes the intersection of sets A and B?
Which property correctly describes the intersection of sets A and B?
What can be concluded if A is a subset of B?
What can be concluded if A is a subset of B?
For sets A and B, which theorem states that A ⊆ B, A ∩ B = A, and A ∪ B = B are equivalent?
For sets A and B, which theorem states that A ⊆ B, A ∩ B = A, and A ∪ B = B are equivalent?
If U is the disjoint union of M and F, which of the following is true?
If U is the disjoint union of M and F, which of the following is true?
Which of the following statements about sets A and B is correct?
Which of the following statements about sets A and B is correct?
What is the intersection of sets A and C where A = {1, 2, 3, 4} and C = {2, 3, 8, 9}?
What is the intersection of sets A and C where A = {1, 2, 3, 4} and C = {2, 3, 8, 9}?
What is the result of applying DeMorgan’s Law to the expression $(A ∪ B)^C$?
What is the result of applying DeMorgan’s Law to the expression $(A ∪ B)^C$?
Which logical equivalence represents the negation of a disjunction?
Which logical equivalence represents the negation of a disjunction?
What defines a finite set?
What defines a finite set?
Which of the following sets is countably infinite?
Which of the following sets is countably infinite?
What is the dual of the equation $(U ∩ A) ∪ (B ∩ A) = A$?
What is the dual of the equation $(U ∩ A) ∪ (B ∩ A) = A$?
How is the notation $n(S)$ used in set theory?
How is the notation $n(S)$ used in set theory?
Which of the following sets is uncountable?
Which of the following sets is uncountable?
What is the notation used to denote the empty set?
What is the notation used to denote the empty set?
What is the conclusion of Lemma 1.6 regarding the union of two finite disjoint sets A and B?
What is the conclusion of Lemma 1.6 regarding the union of two finite disjoint sets A and B?
When applying the Inclusion-Exclusion Principle to finite sets A and B, what adjustment is made to avoid overcounting?
When applying the Inclusion-Exclusion Principle to finite sets A and B, what adjustment is made to avoid overcounting?
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?
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?
If set A contains 25 elements and overlaps with set B containing 10 elements, what represents n(A extbackslash B)?
If set A contains 25 elements and overlaps with set B containing 10 elements, what represents n(A extbackslash B)?
How can the count of the complement of a finite subset A within a universal set U be determined?
How can the count of the complement of a finite subset A within a universal set U be determined?
What is a significant property of the sets A and B stated in Lemma 1.6?
What is a significant property of the sets A and B stated in Lemma 1.6?
In the context of the Inclusion–Exclusion Principle, which statement is correct?
In the context of the Inclusion–Exclusion Principle, which statement is correct?
What is the relationship between the sets A, B, and their intersection in the Inclusion–Exclusion Principle?
What is the relationship between the sets A, B, and their intersection in the Inclusion–Exclusion Principle?
What does the complement of a set A, denoted by AC, represent?
What does the complement of a set A, denoted by AC, represent?
Which of the following notations is used for the relative complement of a set B with respect to A?
Which of the following notations is used for the relative complement of a set B with respect to A?
How is the symmetric difference of two sets A and B denoted?
How is the symmetric difference of two sets A and B denoted?
If A = {1, 2, 3, 4} and B = {3, 4, 5, 6, 7}, what is the result of A extbackslash B?
If A = {1, 2, 3, 4} and B = {3, 4, 5, 6, 7}, what is the result of A extbackslash B?
In the expression A ⊕ B = (A ∪ B) extbackslash(A ∩ B), what does A ∪ B represent?
In the expression A ⊕ B = (A ∪ B) extbackslash(A ∩ B), what does A ∪ B represent?
What is the result of the symmetric difference A ⊕ C if A = {1, 2, 3, 4} and C = {2, 3, 8, 9}?
What is the result of the symmetric difference A ⊕ C if A = {1, 2, 3, 4} and C = {2, 3, 8, 9}?
Which mathematical operation is used to represent the fundamental product of sets A1, A2, ..., An?
Which mathematical operation is used to represent the fundamental product of sets A1, A2, ..., An?
Which of the following correctly represents the relative complement of A with respect to the universal set U?
Which of the following correctly represents the relative complement of A with respect to the universal set U?
Flashcards
Universal Set (U)
Universal Set (U)
A set that includes all other sets being considered in a particular context. It is the largest set.
Empty Set (∅)
Empty Set (∅)
A set with no elements.
Subset
Subset
A set is a subset of another set if every element of the first set is also an element of the second set.
Disjoint Sets
Disjoint Sets
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Venn Diagram
Venn Diagram
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Venn Diagram: Overlapping Sets
Venn Diagram: Overlapping Sets
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Venn Diagram: Disjoint Sets
Venn Diagram: Disjoint Sets
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Venn Diagram: Subset
Venn Diagram: Subset
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What is a set?
What is a set?
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What does '∈' mean?
What does '∈' mean?
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How are sets specified?
How are sets specified?
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When are sets equal?
When are sets equal?
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What is a subset?
What is a subset?
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What is the empty set?
What is the empty set?
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What does '∈/' mean?
What does '∈/' mean?
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How are sets combined?
How are sets combined?
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Disjoint Union
Disjoint Union
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Intersection Subset
Intersection Subset
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Set A is a Subset of the Union
Set A is a Subset of the Union
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Set Inclusion and Intersection
Set Inclusion and Intersection
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Set Inclusion and Union
Set Inclusion and Union
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Proper Subset
Proper Subset
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Complement of a set
Complement of a set
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Difference (A - B)
Difference (A - B)
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Symmetric Difference (A ⊕ B)
Symmetric Difference (A ⊕ B)
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Fundamental Product
Fundamental Product
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Union of sets
Union of sets
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Intersection of sets
Intersection of sets
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Universal Set
Universal Set
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Set Identity
Set Identity
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Intersection of two sets
Intersection of two sets
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Dual of a Set Algebra Equation
Dual of a Set Algebra Equation
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Principle of Duality
Principle of Duality
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Finite Set
Finite Set
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Infinite Set
Infinite Set
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Countable Set
Countable Set
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Uncountable Set
Uncountable Set
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Cardinality of a Set
Cardinality of a Set
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Lemma 1.6: Disjoint Union
Lemma 1.6: Disjoint Union
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Corollary 1.7: Set Difference
Corollary 1.7: Set Difference
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Corollary 1.8: Complement
Corollary 1.8: Complement
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Inclusion-Exclusion Principle
Inclusion-Exclusion Principle
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Corollary 1.10: Three Sets Inclusion-Exclusion
Corollary 1.10: Three Sets Inclusion-Exclusion
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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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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.
