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Questions and Answers
What is the property of a set where the order of elements does not matter?
Which set notation represents a set by describing the properties of its elements?
What is the set of all elements that are in A or in B or in both, denoted by?
A set with no elements is denoted by?
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What is the relationship between two sets A and B, denoted by A ⊆ B?
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What is the set of all elements that are not in A, denoted by?
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What is the property of a set that ensures each element is unique and cannot be repeated?
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A set A is a _______ of a set B, denoted by A ⊂ B, if A is a subset of B and A is not equal to B.
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Study Notes
Definition
A set is a collection of unique objects, known as elements or members, that can be anything (numbers, letters, people, etc.).
Properties
- Uniqueness: Each element in a set is unique and cannot be repeated.
- Order: The order of elements in a set does not matter.
- Finite or Infinite: A set can be either finite (having a limited number of elements) or infinite (having an unlimited number of elements).
Notation
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Roster Form: A set is represented by listing its elements, separated by commas, and enclosed in curly braces { }.
- Example: {1, 2, 3, 4, 5}
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Set Builder Notation: A set is represented by describing the properties of its elements.
- Example: {x | x is a natural number less than 10}
Operations
- Union: The union of two sets A and B, denoted by A ∪ B, is the set of all elements that are in A or in B or in both.
- Intersection: The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that are in both A and B.
- Complement: The complement of a set A, denoted by A', is the set of all elements that are not in A.
- Difference: The difference of two sets A and B, denoted by A \ B, is the set of all elements that are in A but not in B.
Types of Sets
- Empty Set: A set with no elements, denoted by ∅.
- Singleton Set: A set with only one element.
- Universal Set: A set that contains all possible elements in a particular context.
Relationships Between Sets
- Subset: A set A is a subset of a set B, denoted by A ⊆ B, if every element of A is also an element of B.
- Proper Subset: A set A is a proper subset of a set B, denoted by A ⊂ B, if A is a subset of B and A is not equal to B.
- Equal Sets: Two sets A and B are equal, denoted by A = B, if they have the same elements.
Definition of a Set
- A set is a collection of unique objects, known as elements or members.
- Elements can be anything (numbers, letters, people, etc.).
Properties of Sets
- Uniqueness: Each element in a set is unique and cannot be repeated.
- Order: The order of elements in a set does not matter.
- Finite or Infinite: A set can be either finite (having a limited number of elements) or infinite (having an unlimited number of elements).
Set Notation
- Roster Form: Lists elements, separated by commas, and enclosed in curly braces { }.
- Set Builder Notation: Describes the properties of its elements.
Set Operations
- Union: A ∪ B is the set of all elements that are in A or in B or in both.
- Intersection: A ∩ B is the set of all elements that are in both A and B.
- Complement: A' is the set of all elements that are not in A.
- Difference: A \ B is the set of all elements that are in A but not in B.
Types of Sets
- Empty Set: A set with no elements, denoted by ∅.
- Singleton Set: A set with only one element.
- Universal Set: A set that contains all possible elements in a particular context.
Relationships Between Sets
- Subset: A ⊆ B if every element of A is also an element of B.
- Proper Subset: A ⊂ B if A is a subset of B and A is not equal to B.
- Equal Sets: A = B if they have the same elements.
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Description
Learn about the definition, properties, and notation of sets in mathematics, including uniqueness, order, and finite or infinite elements.
