Podcast
Questions and Answers
Which of the following collections would be considered a well-defined set?
Which of the following collections would be considered a well-defined set?
- The set of all difficult math problems.
- The set of all attractive people living in London
- The set of the best songs ever recorded.
- The set of all students enrolled in exactly one IT course at Harvard University in 2024 (correct)
If set A = {2, 4, 6, 8} and set B = {even numbers less than 10}, which statement is correct?
If set A = {2, 4, 6, 8} and set B = {even numbers less than 10}, which statement is correct?
- Set A and Set B both contain an infinite amount of elements
- Set A is not equal to set B because they are described differently.
- Set A is a subset of set B, but not equal.
- Set A is equal to set B because they contain exactly the same elements. (correct)
Given a universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and a set A = {2, 4, 6, 8}, which elements of U are NOT elements of A?
Given a universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and a set A = {2, 4, 6, 8}, which elements of U are NOT elements of A?
- {1, 3, 5, 7, 9} (correct)
- {9, 10}
- {2, 4, 6, 8}
- {1, 2, 3, 4, 5}
Which of the following sets is an example of an infinite set?
Which of the following sets is an example of an infinite set?
If x ∈ A means 'x is an element of A,' and set A = {1, 3, 5, 7, 9}, which of the following statements is true?
If x ∈ A means 'x is an element of A,' and set A = {1, 3, 5, 7, 9}, which of the following statements is true?
Which of the following pairs of sets, A and B, are equal?
Which of the following pairs of sets, A and B, are equal?
Given the universal set U = {1, 2, 3, ..., 10}, which of the following sets is a subset of U?
Given the universal set U = {1, 2, 3, ..., 10}, which of the following sets is a subset of U?
If set A is a proper subset of set B, which of the following statements must be true?
If set A is a proper subset of set B, which of the following statements must be true?
Which of the following sets is equivalent to {x | x is an even number between 1 and 9}?
Which of the following sets is equivalent to {x | x is an even number between 1 and 9}?
Given A = {1, 2, 3} and B = {2, 3, 4}, which of the following statements is true?
Given A = {1, 2, 3} and B = {2, 3, 4}, which of the following statements is true?
If U is the universal set of all integers, which of the following sets is NOT a subset of U?
If U is the universal set of all integers, which of the following sets is NOT a subset of U?
Which of the following statements accurately describes a proper subset?
Which of the following statements accurately describes a proper subset?
Consider A = {a, b, c}. Which of the following sets would make A a proper subset?
Consider A = {a, b, c}. Which of the following sets would make A a proper subset?
Given set X = {apple, banana, cherry}, which of the following sets is a proper subset of X?
Given set X = {apple, banana, cherry}, which of the following sets is a proper subset of X?
How does set theory establish a foundation for data structures represented in computer science?
How does set theory establish a foundation for data structures represented in computer science?
Set A = {2, 4, 6, 8}. How many subsets does set A have?
Set A = {2, 4, 6, 8}. How many subsets does set A have?
If set Y = {red, green, blue}, how many proper subsets does Y have?
If set Y = {red, green, blue}, how many proper subsets does Y have?
Which of the following is NOT a proper subset of {1, 2, 3}?
Which of the following is NOT a proper subset of {1, 2, 3}?
Given A = {x, y, z}, what is the number of proper subsets of A?
Given A = {x, y, z}, what is the number of proper subsets of A?
Consider the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and set A = {2, 4, 6, 8}. Which visual representation best describes the relationship between U and A?
Consider the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and set A = {2, 4, 6, 8}. Which visual representation best describes the relationship between U and A?
A set C has 5 elements. How many more subsets does C have compared to its number of proper subsets?
A set C has 5 elements. How many more subsets does C have compared to its number of proper subsets?
Given set A = {a, b, c, d, e} and set B = {c, e, f, g}, what is the result of A - B?
Given set A = {a, b, c, d, e} and set B = {c, e, f, g}, what is the result of A - B?
If set X = {2, 4, 6, 8} and set Y = {1, 2, 3, 4, 5}, what is the symmetric difference of X and Y (X ⊕ Y)?
If set X = {2, 4, 6, 8} and set Y = {1, 2, 3, 4, 5}, what is the symmetric difference of X and Y (X ⊕ Y)?
In a survey, it's found that out of 100 people, 60 like apples, 50 like bananas, and 20 like both. How many people like only apples?
In a survey, it's found that out of 100 people, 60 like apples, 50 like bananas, and 20 like both. How many people like only apples?
Suppose A = {1, 2, 3}, B = {2, 4}, and C = {3, 4, 5}. What is A ⊕ (B ∪ C)?
Suppose A = {1, 2, 3}, B = {2, 4}, and C = {3, 4, 5}. What is A ⊕ (B ∪ C)?
A school club has 30 members. 15 are in the math club, 12 are in the science club, and 5 are in both. How many students are in neither the math nor the science club?
A school club has 30 members. 15 are in the math club, 12 are in the science club, and 5 are in both. How many students are in neither the math nor the science club?
Consider three sets: A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, and C = {4, 6, 7, 8}. What elements are in (A ∩ B) ⊕ C?
Consider three sets: A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, and C = {4, 6, 7, 8}. What elements are in (A ∩ B) ⊕ C?
In a group of 50 people, 30 own a car, 25 own a bike, and 10 own neither. How many people own both a car and a bike?
In a group of 50 people, 30 own a car, 25 own a bike, and 10 own neither. How many people own both a car and a bike?
Given U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 5, 7, 9}, and B = {2, 4, 6, 8, 10}. Determine $(A \cup B)'$.
Given U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 5, 7, 9}, and B = {2, 4, 6, 8, 10}. Determine $(A \cup B)'$.
Given the universal set $U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ and set $A = {2, 4, 6, 8, 10}$, what is the complement of set A, denoted as A'?
Given the universal set $U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ and set $A = {2, 4, 6, 8, 10}$, what is the complement of set A, denoted as A'?
If $X = {a, b, c, d, e}$ and $Y = {c, d, e, f, g}$, which of the following represents the intersection of sets X and Y (X ∩ Y)?
If $X = {a, b, c, d, e}$ and $Y = {c, d, e, f, g}$, which of the following represents the intersection of sets X and Y (X ∩ Y)?
Let $P = {1, 3, 5}$ and $Q = {2, 4, 6}$. What is the union of sets P and Q (P ∪ Q)?
Let $P = {1, 3, 5}$ and $Q = {2, 4, 6}$. What is the union of sets P and Q (P ∪ Q)?
Given $E = {2, 4, 6, 8}$ and $F = {4, 8, 12, 16}$, what is the set difference E - F?
Given $E = {2, 4, 6, 8}$ and $F = {4, 8, 12, 16}$, what is the set difference E - F?
Consider the universal set $U = {1, 2, 3, 4, 5, 6, 7, 8}$, $A = {1, 2, 3, 4}$, and $B = {3, 4, 5, 6}$. What is $(A ∪ B)'$?
Consider the universal set $U = {1, 2, 3, 4, 5, 6, 7, 8}$, $A = {1, 2, 3, 4}$, and $B = {3, 4, 5, 6}$. What is $(A ∪ B)'$?
If the universal set $U$ is all positive integers, $A$ is the set of even numbers, and $B$ is the set of multiples of 3, what does $A ∩ B$ represent?
If the universal set $U$ is all positive integers, $A$ is the set of even numbers, and $B$ is the set of multiples of 3, what does $A ∩ B$ represent?
Let $U = {a, b, c, d, e, f}$, $X = {a, c, e}$, and $Y = {b, c, f}$. Determine the result of $(X - Y) ∪ (Y - X)$.
Let $U = {a, b, c, d, e, f}$, $X = {a, c, e}$, and $Y = {b, c, f}$. Determine the result of $(X - Y) ∪ (Y - X)$.
Suppose $U$ is the set of all students in a university, $A$ is the set of students taking mathematics, and $B$ is the set of students taking computer science. What does $A - B$ represent?
Suppose $U$ is the set of all students in a university, $A$ is the set of students taking mathematics, and $B$ is the set of students taking computer science. What does $A - B$ represent?
Flashcards
What is a Set?
What is a Set?
A well-defined, unordered collection of objects (elements/members).
Roster Method
Roster Method
Listing all elements (e.g., S = {0, 1, 2, 3}).
Set Builder Method
Set Builder Method
Describing the set's contents (e.g., S = {whole numbers less than 10}).
Universal Set
Universal Set
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Finite Set
Finite Set
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Equal Sets
Equal Sets
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Subset
Subset
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Subset Symbolism
Subset Symbolism
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A = {4, 5, 6} is a subset of set U
A = {4, 5, 6} is a subset of set U
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A = {..., -8, -4, 0, 4, 8,...} is a subset of set U
A = {..., -8, -4, 0, 4, 8,...} is a subset of set U
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Proper Subset
Proper Subset
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A = {x | x is a rational number} is a subset of set U
A = {x | x is a rational number} is a subset of set U
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A = {3a, 2x, -1y} is not a subset of set U
A = {3a, 2x, -1y} is not a subset of set U
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Set Difference (A - B)
Set Difference (A - B)
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Symmetric Difference (A ⊕ B)
Symmetric Difference (A ⊕ B)
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Venn Diagram
Venn Diagram
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Intersection (A ∩ B)
Intersection (A ∩ B)
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Universal Set (U)
Universal Set (U)
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Null Set/Empty Set
Null Set/Empty Set
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Intersection of Multiple Sets
Intersection of Multiple Sets
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Union of Sets
Union of Sets
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Proper Subset Definition
Proper Subset Definition
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Number of Subsets
Number of Subsets
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Number of Proper Subsets
Number of Proper Subsets
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A ⊆ B
A ⊆ B
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A≠B
A≠B
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Set Complement (S’ or Sc)
Set Complement (S’ or Sc)
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Set Intersection (∩)
Set Intersection (∩)
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Set Union (∪)
Set Union (∪)
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Algebra of Sets
Algebra of Sets
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Set Theory
Set Theory
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Study Notes
- A set is a well-defined and unordered collection of objects, referred to as elements or members.
- Sets are denoted by uppercase letters.
Determining if a Set is Well-Defined
- A set is well-defined if its contents can be clearly and unambiguously determined.
- The set of all official James Bond films made by EON Productions is well-defined.
- The set of the best TV shows of all time is not well-defined.
- The set of the top 10 selling recording artists of 2016 is well-defined.
- The set of great rap artists is not well-defined.
Set Notation Methods
- Roster Method: Lists all elements, e.g., S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
- Descriptive Method: Describes the set, e.g., S = {whole numbers less than 10}.
Universal Set
- Includes of all elements relevant to a particular discussion or problem.
- Example: U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, which can be divided into EVEN and ODD subsets.
Element Notation
- x ∈ S denotes that x is an element of set S.
- x ∉ S denotes that x is not an element of set S.
Finite vs. Infinite Sets
- Finite Set: Contains a countable number of elements.
- A = {whole numbers less than 5} = {0, 1, 2, 3, 4} has 5 elements.
- B = {letters of the alphabet} has 26 elements
- Infinite Set: Contains a non-countable number of elements.
- X = {even whole numbers} = {2, 4, 6, 8, ...}
- Y = {multiples of 100} = {100, 200, 300, ...}
- The three dots (...) indicate the list continues infinitely.
Set Equality
- Two sets are identical if and only if they contain exactly the same elements.
- A = {9, 2, 7, -3} and B = {7, 9, -3, 2} are equal.
Subset
- A set contained within a larger or equal set.
- If U is the universal set of all numbers, then A is a subset of U if every element in A is also in U.
- Symbolically, A ⊆ U if ∀x [x ∈ A → x ∈ U].
Proper Subset
- A subset that is not equal to the set it belongs to.
- Symbolically, A ⊂ B if A ⊆ B and A ≠B.
Number of Subsets
- The number of subsets of a set with n elements is 2n.
- A = {dog, cat} has 22 = 4 subsets.
- B = {dog, cat, bird, fish} has 24 = 16 subsets.
Number of Proper Subsets
- The number of proper subsets of a set with n elements is 2n - 1.
- A = {dog, cat} has 22 - 1 = 3 proper subsets.
- B = {dog, cat, bird, fish} has 24 - 1 = 15 proper subsets.
Venn Diagrams
- Visually represent sets and their relationships.
- The universal set (U) is typically drawn as a large rectangle.
- Other sets are represented by circles within this rectangle.
Set Complement
- The complement of a set S includes all elements of the universal set U that are not in S, denoted as S' or Sc.
- If U = {letters of the alphabet} and V = {vowels}, then V' = {not vowels or consonants}.
Set Intersection
- The intersection of two sets A and B (A ∩ B) includes all elements belonging to both A and B.
- If A = {girls} and B = {adults}, then A ∩ B = {adults who are girls}.
- If C = {0, 1, 2, 3,..., 98, 99, 100} and D = {50, 100, 150, 200, 250,...}, then C ∩ D = {50, 100}.
Set Union
- The union of two sets A and B (A ∪ B) includes all elements belonging to either A or B.
- If A = {even numbers} and B = {odd numbers}, then A ∪ B = {even and odd numbers}.
- If C = {1, 2, 3, 4, 5} and D = {6, 7, 8, 9, 10}, then C ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
Set Difference
- The difference between set A and set B (A - B) is composed of elements of set A that are not in set B.
- If A = {1, 2, 3,..., 9, 10} and B = {2, 4, 6, 8, 10}, then A - B = {1, 3, 5, 7, 9}.
- B - A = {} (null set), all elements of set B are in set A
Symmetric Difference
- The symmetric difference between two sets is the set of all elements present in either of the sets but not in their intersection.
- If A = {3, 6, 9} and B = {2, 4, 6, 8, 10}, then A ⊕ B = {2, 3, 4, 8, 9, 10}.
- The element {6} is in both sets A and B, and thus not in their symmetric difference.
- A ∩ B = {6}, so (A ∩ B)' = A ⊕ B = {2, 3, 4, 8, 9, 10}.
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Description
Explore set theory concepts. This includes definitions of sets, universal sets, subsets, and equivalent sets. Practice identifying well-defined, infinite, and equal sets through examples.
