Summary
Highlights
A set is a collection of distinct objects or elements that share similar properties. Membership in a set is unambiguous. Sets can be named, and elements are identified using the 'element of' symbol (∈) or 'not an element of' symbol (∉). Set builder notation is a shorthand for describing sets, specifying a variable and a predicate it must satisfy. When dealing with numerical sets, it's important to declare the starting set explicitly.
Two sets are equal if they contain the same elements, regardless of order or repetition. The size or cardinality of a set is the number of elements it contains, denoted by two vertical lines (||). Infinite sets can have a cardinality of infinity.
A set A is a subset of set B if all elements of A are also elements of B, denoted by ⊆. It's crucial to distinguish between elements and subsets; subsets are always sets themselves. A set is always a subset of itself. If A is a subset of B but A is not equal to B, then A is a proper subset of B, implying B contains elements not in A. This is sometimes denoted by ⊂. If A ⊆ B and B ⊆ C, then A ⊆ C.
The empty set (∅) is a special set containing no elements. It is a subset of any set, and it is unique.
The union of two sets A and B (A ∪ B) is a new set containing all elements from A or B. The intersection of two sets A and B (A ∩ B) is a set containing only the elements common to both A and B. If there are no common elements, the intersection is the empty set. Examples are provided with numerical sets and odd/even numbers.
Properties of unions include: A ∪ ∅ = A, A ∪ A = A, if A ⊆ B then A ∪ B = B, and A ∪ B = B ∪ A (commutative property). Unions are also associative: (A ∪ B) ∪ C = A ∪ (B ∪ C). Properties of intersections include: A ∩ ∅ = ∅, A ∩ A = A, if A ⊆ B then A ∩ B = A, and A ∩ B = B ∩ A (commutative property). Intersections are also associative. The cardinality of A ∪ B is |A| + |B| - |A ∩ B|. The distributive property applies to both unions and intersections, allowing for manipulation of logical statements, as demonstrated with examples about cycling and shirts.
A formal method to prove set equality, such as the distributive property, involves showing that each set is a subset of the other. This is done by taking an arbitrary element and demonstrating its membership in both sides of the equation.
The set theoretic difference of A and B (A \ B) contains all elements in A that are not in B. If B is a subset of A, this difference is called the complement of B with respect to A (Bᶜ or C(B)). The universal set (U) encompasses all relevant elements for a given topic. The complement of A (Aᶜ) refers to all elements in the universal set that are not in A, effectively negating the predicate defining A.
Properties of complements include: ∅ᶜ = U, Uᶜ = ∅, (Aᶜ)ᶜ = A. If A ⊆ B, then Bᶜ ⊆ Aᶜ. De Morgan's Laws state: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ and (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ. These laws demonstrate how complements interact with unions and intersections, illustrating the De Morgan duality principle where interchanging union and intersection symbols in an identity results in another valid identity.
A power set P(A) of a set A contains all possible subsets of A. An indexed family of sets refers to a collection of sets where each set is identified by an index, often a number, and written as a subscript (e.g., Aᵢ).
Russell's Paradox highlights a fundamental problem in 'naive' set theory, raising questions about sets that contain themselves or don't contain themselves, leading to logical contradictions. Axiomatic set theory aims to resolve these paradoxes by providing rigorous definitions and axioms for what constitutes a valid set, guiding mathematical reasoning.