Sets and Set Operations

Sets

• A set is a collection of unique objects, known as elements or members, that can be anything (numbers, people, letters, etc.)
• Sets are denoted using curly braces {} and elements are separated by commas
• Example: {1, 2, 3, 4, 5} is a set of integers
• Sets can be finite (limited number of elements) or infinite (unlimited number of elements)

Set Operations

• Union: The union of two sets A and B, denoted as A ∪ B, is the set of all elements in A or B or both
• Intersection: The intersection of two sets A and B, denoted as A ∩ B, is the set of all elements common to A and B
• Difference: The difference of two sets A and B, denoted as A - B, is the set of all elements in A but not in B
• Complement: The complement of a set A, denoted as A', is the set of all elements not in A

Types of Sets

• Empty Set: A set with no elements, denoted as {}
• Singleton Set: A set with only one element, denoted as {a}
• Universal Set: A set that contains all elements in a particular context, denoted as U

Functions

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range)
• Functions can be represented as:
  • Arrow notation: f: A → B
  • Set notation: {(a, b) | a ∈ A, b ∈ B}
• A function is said to be injective (one-to-one) if every element in the range corresponds to at most one element in the domain
• A function is said to be surjective (onto) if every element in the range corresponds to at least one element in the domain
• A function is said to be bijective (one-to-one correspondence) if it is both injective and surjective

Sets

• A set is a collection of unique objects called elements or members.
• Sets can be represented using curly braces {} and elements are separated by commas.
• Example: {1, 2, 3, 4, 5} is a set of integers.
• Sets can be either finite (limited number of elements) or infinite (unlimited number of elements).

Set Operations

• The union of two sets A and B, denoted as A ∪ B, is the set of all elements in A or B or both.
• The intersection of two sets A and B, denoted as A ∩ B, is the set of all elements common to A and B.
• The difference of two sets A and B, denoted as A - B, is the set of all elements in A but not in B.
• The complement of a set A, denoted as A', is the set of all elements not in A.

Types of Sets

• An empty set is a set with no elements, denoted as {}.
• A singleton set is a set with only one element, denoted as {a}.
• A universal set is a set that contains all elements in a particular context, denoted as U.

Functions

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
• Functions can be represented using arrow notation as f: A → B or set notation as {(a, b) | a ∈ A, b ∈ B}.
• A function is injective (one-to-one) if every element in the range corresponds to at most one element in the domain.
• A function is surjective (onto) if every element in the range corresponds to at least one element in the domain.
• A function is bijective (one-to-one correspondence) if it is both injective and surjective.