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.

Learn about sets, including what they are, how they are denoted, and their properties. Also, discover set operations like union and intersection.