Sets in Math

Study Notes

Definition: A set is a collection of unique objects, known as elements or members, that can be anything (numbers, letters, people, etc.).

Key Concepts:

Elements: Individual objects within a set, denoted by lowercase letters (e.g., a, b, c).
Set Notation: Sets are typically denoted using curly brackets {} and elements are separated by commas.
Empty Set: A set with no elements, denoted by ∅ or {}.
Universal Set: A set that contains all elements under consideration, denoted by U.

Operations:

Union: The combination of two or more sets, denoted by ∪. Example: A ∪ B = {elements in A or B or both}.
Intersection: The set of elements common to two or more sets, denoted by ∩. Example: A ∩ B = {elements in both A and B}.
Difference: The set of elements in one set but not in another, denoted by -. Example: A - B = {elements in A but not in B}.
Complement: The set of elements not in a given set, denoted by '. Example: A' = {elements in U but not in A}.

Properties:

Commutative Property: The order of sets does not change the result of union and intersection operations. Example: A ∪ B = B ∪ A.
Associative Property: The order in which sets are combined does not change the result of union and intersection operations. Example: (A ∪ B) ∪ C = A ∪ (B ∪ C).
Distributive Property: The union and intersection operations can be distributed over each other. Example: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

Types of Sets:

Finite Set: A set with a fixed number of elements.
Infinite Set: A set with an infinite number of elements.
Subset: A set contained within another set. Example: A is a subset of B if every element of A is also in B.
Power Set: The set of all possible subsets of a given set.

The union of two or more sets is denoted by ∪ and contains all elements in at least one set.
The intersection of two or more sets is denoted by ∩ and contains all elements common to all sets.
The difference of two sets is denoted by - and contains all elements in the first set but not in the second.
The complement of a set is denoted by ' and contains all elements in the universal set but not in the given set.

The commutative property states that the order of sets does not change the result of union and intersection operations.
The associative property states that the order in which sets are combined does not change the result of union and intersection operations.
The distributive property states that the union and intersection operations can be distributed over each other.