What Is a Set?
A set is a well-defined collection of distinct objects, called elements or members. Sets are one of the most fundamental concepts in mathematics.
Notation:
{1, 2, 3} - Set enumeration
{x | x > 0} - Set builder notation
|A| - Cardinality (size) of set A
a ∈ A - "a is an element of A"
a ∉ A - "a is not an element of A"Types of Sets
Empty Set (∅): The set with no elements: {}
Universal Set (U): The set of all elements under consideration
Singleton Set: A set with exactly one element: {a}
Finite Set: A set with a limited number of elements
Infinite Set: A set with unlimited elements (e.g., ℕ, ℤ)Subsets
A set A is a subset of B (written A ⊆ B) if every element of A is also an element of B. Every set is a subset of itself, and the empty set is a subset of every set.
A ⊆ B iff ∀x (x ∈ A → x ∈ B)
A ⊂ B (A is a proper subset of B: A ⊆ B and A ≠ B)
Power Set: P(A) = the set of all subsets of A
If |A| = n, then |P(A)| = 2ⁿ