Podcast
Questions and Answers
Which of the following mathematicians is credited with introducing the concept of sets?
Which of the following mathematicians is credited with introducing the concept of sets?
- Georg Cantor (correct)
- Alan Turing
- Euclid
- Pythagoras
The set of all real numbers is denoted by the symbol ℤ.
The set of all real numbers is denoted by the symbol ℤ.
False (B)
What is the term for a set that contains all elements of related sets under consideration?
What is the term for a set that contains all elements of related sets under consideration?
Universal Set
A __________ is a collection of related and well-defined objects called elements.
A __________ is a collection of related and well-defined objects called elements.
Match the set notation examples with their descriptions:
Match the set notation examples with their descriptions:
Which of the sets below is equivalent to $D = {x | x \in ℕ, 3 < x < 7}$?
Which of the sets below is equivalent to $D = {x | x \in ℕ, 3 < x < 7}$?
If set A is a subset of set B, then set B must be a subset of set A.
If set A is a subset of set B, then set B must be a subset of set A.
Express the following set in set-builder notation: The set of all even numbers greater than 0.
Express the following set in set-builder notation: The set of all even numbers greater than 0.
Which of the following is NOT typically a topic covered within the course 'GE Math 1a', according to the course outline?
Which of the following is NOT typically a topic covered within the course 'GE Math 1a', according to the course outline?
When writing a set using __________, elements are listed and separated by commas within curly braces.
When writing a set using __________, elements are listed and separated by commas within curly braces.
Which of the following statements accurately describes an empty set?
Which of the following statements accurately describes an empty set?
If set A = {a, b, c, d}, then n(A) = 3.
If set A = {a, b, c, d}, then n(A) = 3.
If A is a subset of B, what must be true about the elements of A in relation to B?
If A is a subset of B, what must be true about the elements of A in relation to B?
If set A has n
elements, then the number of subsets of A is given by the formula 2 to the power of ______.
If set A has n
elements, then the number of subsets of A is given by the formula 2 to the power of ______.
Which of the following conditions must be true for set A to be considered a proper subset of set B?
Which of the following conditions must be true for set A to be considered a proper subset of set B?
The power set of a set S includes only the non-empty subsets of S.
The power set of a set S includes only the non-empty subsets of S.
Given set X = {1, 2, 3}, which of the following represents the power set P(X) correctly?
Given set X = {1, 2, 3}, which of the following represents the power set P(X) correctly?
If set A has 3 elements, how many elements are in its power set P(A)?
If set A has 3 elements, how many elements are in its power set P(A)?
A subset that is not a proper subset must be ______ to the original set.
A subset that is not a proper subset must be ______ to the original set.
Which of the following sets is equal to $X = {x ∈ ℕ | 0 < x < 5}$?
Which of the following sets is equal to $X = {x ∈ ℕ | 0 < x < 5}$?
Given a set $S = {a}$, the statement ${a} \subseteq P(S)$ is true.
Given a set $S = {a}$, the statement ${a} \subseteq P(S)$ is true.
For any set, the intersection of that set with its complement results in the universal set.
For any set, the intersection of that set with its complement results in the universal set.
If set $A = {1, 2, 3}$ and set $B = {3, 4, 5}$, what is $A \cup B$?
If set $A = {1, 2, 3}$ and set $B = {3, 4, 5}$, what is $A \cup B$?
Given the universal set $U = {1, 2, 3, 4, 5}$ and the set $A = {2, 4}$, what is $A'$?
Given the universal set $U = {1, 2, 3, 4, 5}$ and the set $A = {2, 4}$, what is $A'$?
If set $A = {a, b}$ and set $B = {1, 2}$, what is the Cartesian product $A \times B$?
If set $A = {a, b}$ and set $B = {1, 2}$, what is the Cartesian product $A \times B$?
If $A={1, 2, 3, 4, 5}$ and $B={4, 5, 6, 7}$, what is $A - B$?
If $A={1, 2, 3, 4, 5}$ and $B={4, 5, 6, 7}$, what is $A - B$?
The ________ of two sets contains all the elements in both sets.
The ________ of two sets contains all the elements in both sets.
Match the set operations with their descriptions:
Match the set operations with their descriptions:
Which of the following statements about set operations is always true?
Which of the following statements about set operations is always true?
Explain the difference between the intersection and the Cartesian product of two sets.
Explain the difference between the intersection and the Cartesian product of two sets.
Flashcards
Set
Set
A collection of related, well-defined objects called elements.
Universal Set
Universal Set
All elements of related sets, including its subsets.
Roster Notation
Roster Notation
A way of listing elements separated by commas within curly braces.
Set-Builder Notation
Set-Builder Notation
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ℝ
ℝ
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ℚ
ℚ
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ℤ
ℤ
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ℕ
ℕ
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𝕎
𝕎
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Element (∈)
Element (∈)
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Power Set P(S)
Power Set P(S)
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Union of Sets
Union of Sets
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Intersection of Sets
Intersection of Sets
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Complement of a Set
Complement of a Set
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Difference of Sets
Difference of Sets
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Cartesian Product
Cartesian Product
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Ordered Pairs
Ordered Pairs
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Empty set
Empty set
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Cardinality of a Set
Cardinality of a Set
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Subset
Subset
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Proper Subset
Proper Subset
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Power Set
Power Set
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Number of Subsets
Number of Subsets
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A ⊆ B
A ⊆ B
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A ⊂ B
A ⊂ B
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Study Notes
- GE Math 1a course covers topics such as the nature and language of mathematics, problem solving, biostatistics, data handling, hypothesis testing, and various statistical tests.
- Class rules include being punctual, respectful, and attentive, as well as prohibiting the use of cellphones.
- Mathematics uses specific symbols, syntax, and rules.
- The learning objectives are to define sets, find subsets, perform set operations, and use Venn Diagrams.
Key Concepts of Sets
- A set is a collection of related, well-defined objects called elements, denoted by ∈.
- Georg Cantor introduced the word set in 1879.
Universal Set
- The universal set contains all elements from related sets and is denoted by U.
- R represents real numbers.
- Q represents rational numbers.
- Z represents integers.
- N represents natural numbers.
- W represents whole numbers.
Writing a Set
- Roster notation lists elements separated by commas. Ex: A = {2, 4, 6, 8}; B = {3, 5, 7, 9}
- Set-builder notation represents properties satisfied by the elements of a set. Ex: X = {x ∈ N|0 < x < 8}
Empty set
- An empty set contains no elements and is denoted as {} or Ø.
- The empty set is always a subset of any set
Cardinality of a set
- The cardinality of set A, denoted by n(A) is the number of elements in A.
- Ex: If A = {2,4,6,8,10}, then n(A) = 5.
Subset
- Subsets are sets within a universal set or another set.
- If A and B are sets, A is a subset of B (A⊆ B) if every element of A is also an element of B.
- A⊆ B means that for all elements x ∈ A, then x ∈ B.
- A⊆ B is read as ""A"" is a subset of ""B"".
- A ⊈ B is read as ""A"" is not a subset of ""B"".
- The formula 2^n determines the number of subsets in a sets with n number of elements.
- If n(A) = 6, then 2^6 = 64 subsets.
Proper subset
- A proper subset is any subset of a set, excluding the set itself.
- A ⊂ B means A is a proper subset of B- every element of A is in B and at least one element of B that is not in A.
Power of a set
- The power of a set is the set of all subsets for any given set, including the empty set.
- For S = {a, i, r}, P(S) is the power of set S: P(S) = { ¢, {a}, {r}, {i}, {a,i}, {a, r}, {i, r}, {i,r,a}}
Operations of sets
- The different operations of sets are:
- Union
- Intersection
- Difference
- Complement
- Cartesian Product
Union
- The union of two or more sets contains ALL the elements in all the sets.
- The union of sets A and B is expressed as A U B.
Intersection
- The intersection of two or more sets contains the common elements in all the sets.
- The intersection of sets A and B is expressed as A ∩ B.
Complement
- The complement of a set contains all of the elements in the universal set U that are not found in the considered set.
- The complement of set A is denoted by A'.
Difference
- The difference of A by B (A – B) is the set containing elements of A excluding elements found in both A and B
- The difference of B by A (B – A) is the set containing elements of B excluding elements found in both A and B.
- The difference of A and B vice versa.
Cartesian Products
- Given sets A and B, the Cartesian product of A and B (A × B) is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
- Symbolically, A×B = {(a, b) where a ∈ A, b ∈ B}.
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Description
This lesson covers the fundamentals of sets, including roster and set-builder notation. Key topics include understanding the universal set and performing set operations. Examples of sets such as Real numbers, Rational Numbers, Integers, Natural numbers, and Whole numbers are also discussed.