User:Ascii/ProofWiki Sampling Notes for Theorems/Set Theory

Sets and Elements

 * 1) Set is Subset of Itself
 * $\forall S: S \subseteq S$
 * 1) Singleton of Element is Subset
 * $x \in S \iff \{x\} \subseteq S$
 * 1) Subset Relation is Transitive
 * $\paren {R \subseteq S} \land \paren {S \subseteq T} \implies R \subseteq T$
 * 1) Equivalence of Definitions of Set Equality
 * $S = T \iff \paren {\forall x: x \in S \iff x \in T}$
 * $S = T \iff S \subseteq T \land T \subseteq S$
 * 1) Set Equals Itself
 * $S = S$
 * 1) Set Inequality
 * $S \ne T \iff \left({S \nsubseteq T}\right) \lor \left({T \nsubseteq S}\right)$
 * 1) Empty Set is Subset of All Sets
 * $\forall S: \O \subseteq S$
 * 1) Empty Set is Unique
 * The empty set $\O$ is unique.
 * 1) Empty Set is Element of Power Set
 * $\forall S: \O \in \mathcal P (S)$
 * 1) Set is Element of its Power Set
 * $S \in \powerset S$
 * 1) Power Set of Empty Set
 * $P \left({\varnothing}\right) = \left\{{\varnothing}\right\}$

Union

 * 1) Union is Idempotent
 * $S \cup S = S$
 * 1) Union is Commutative
 * $S \cup T = T \cup S$
 * 1) Union is Associative
 * $(S \cup T) \cup R = S \cup (T \cup R)$
 * 1) Union with Empty Set
 * $S \cup \O = S$
 * 1) Set is Subset of Union
 * $S \subseteq S \cup T$
 * 1) Set Union Preserves Subsets
 * $A \subseteq B, \ S \subseteq T \implies A \cup S \subseteq B \cup T$
 * 1) Union is Smallest Superset
 * $\paren {S_1 \subseteq T} \land \paren {S_2 \subseteq T} \iff \paren {S_1 \cup S_2} \subseteq T$
 * 1) Union of Subsets is Subset
 * $\left({S_1 \subseteq T}\right) \land \left({S_2 \subseteq T}\right) \implies \left({S_1 \cup S_2}\right) \subseteq T$
 * 1) Union with Superset is Superset
 * $S \subseteq T \iff S \cup T = T$
 * 1) Union Distributes over Union
 * $\forall A, B, C: \left({A \cup B}\right) \cup \left({A \cup C}\right) = A \cup B \cup C = \left({A \cup C}\right) \cup \left({B \cup C}\right)$
 * 1) Power Set is Closed under Union

Intersection

 * 1) Intersection is Idempotent
 * $S \cap S = S$
 * 1) Intersection is Commutative
 * $S \cap T = T \cap S$
 * 1) Intersection is Associative
 * $A \cap \paren {B \cap C} = \paren {A \cap B} \cap C$
 * 1) Intersection is Subset
 * $S \cap T \subseteq S$
 * 1) Intersection with Empty Set
 * $S \cap \O = \O$
 * 1) Intersection of Subsets is Subset/Set of Sets
 * Let $T$ be a set and $\mathbb S$ be a non-empty set of sets.
 * Suppose that for each $S \in \mathbb S$: $S \subseteq T$
 * Then $\bigcap \mathbb S \subseteq T$
 * 1) Intersection with Subset is Subset‎
 * $S \subseteq T \iff S \cap T = S$
 * 1) Intersection Distributes over Intersection
 * $\forall A, B, C: \left({A \cap B}\right) \cap \left({A \cap C}\right) = A \cap B \cap C = \left({A \cap C}\right) \cap \left({B \cap C}\right)$
 * 1) Power Set is Closed under Intersection

Union and Intersection

 * 1) Intersection is Subset of Union
 * $S \cap T \subseteq S \cup T$
 * 1) Intersection Distributes over Union
 * $R \cap \paren {S \cup T} = \paren {R \cap S} \cup \paren {R \cap T}$
 * 1) Union Distributes over Intersection
 * $R \cup \paren {S \cap T} = \paren {R \cup S} \cap \paren {R \cup T}$
 * 1) Absorption Laws (Set Theory)/Intersection with Union
 * $S \cap \paren {S \cup T} = S$
 * 1) Absorption Laws (Set Theory)/Union with Intersection
 * $S \cup \paren {S \cap T} = S$
 * 1) Union equals Intersection iff Sets are Equal
 * $\left({S \cup T = S}\right) \land \left({S \cap T = S}\right) \iff S = T$

Set Difference

 * 1) Set Difference is Subset
 * $S \setminus T \subseteq S$
 * 1) Set Difference with Empty Set is Self
 * $S \setminus \O = S$
 * 1) Set Difference with Superset is Empty Set
 * $S \subseteq T \iff S \setminus T = \O$
 * 1) Set Difference with Self is Empty Set
 * $S \setminus S = \O$
 * 1) Set Difference Equals First Set iff Empty Intersection
 * $S \setminus T = S \iff S \cap T = \O$
 * 1) Equal Set Differences iff Equal Intersections
 * $R \setminus S = R \setminus T \iff R \cap S = R \cap T$
 * 1) Set Difference Union Second Set is Union
 * $\left({S \setminus T}\right) \cup T = S \cup T$
 * 1) Set Difference Union First Set is First Set
 * $\paren {S \setminus T} \cup S = S$
 * 1) Set Difference with Union is Set Difference
 * $\left({S \cup T}\right) \setminus T = S \setminus T$
 * 1) Intersection with Set Difference is Set Difference with Intersection
 * $\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus S$
 * 1) Set Difference Intersection with Second Set is Empty Set
 * $\left({S \setminus T}\right) \cap T = \varnothing$
 * 1) De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection
 * $S \setminus \paren {T_1 \cap T_2} = \paren {S \setminus T_1} \cup \paren {S \setminus T_2}$
 * 1) De Morgan's Laws (Set Theory)/Set Difference/Difference with Union
 * $S \setminus \paren {T_1 \cup T_2} = \paren {S \setminus T_1} \cap \paren {S \setminus T_2}$
 * 1) Set Difference with Union
 * $R \setminus \left({S \cup T}\right) = \left({R \cup T}\right) \setminus \left({S \cup T}\right) = \left({R \setminus S}\right) \setminus T = \left({R \setminus T}\right) \setminus S$
 * 1) Set Difference with Set Difference is Union of Set Difference with Intersection
 * $R \setminus \left({S \setminus T}\right) = \left({R \setminus S}\right) \cup \left({R \cap T}\right)$
 * 1) Set Difference Union Intersection
 * $S = \left({S \setminus T}\right) \cup \left({S \cap T}\right)$
 * 1) Set Difference of Intersection with Set is Empty Set
 * $\left({S \cap T}\right) \setminus S = \varnothing$
 * $\left({S \cap T}\right) \setminus T = \varnothing$
 * 1) Set Difference is Anticommutative
 * $S = T \iff S \setminus T = T \setminus S = \varnothing$

Relative Complement

 * 1) Relative Complement of Empty Set
 * $\complement_S \left({\varnothing}\right) = S$
 * 1) Relative Complement with Self is Empty Set
 * $\complement_S \left({S}\right) = \varnothing$
 * 1) Relative Complement of Relative Complement
 * $\relcomp S {\relcomp S T} = T$
 * 1) Intersection with Relative Complement is Empty
 * $T \cap \complement_S \left({T}\right) = \varnothing$
 * 1) Union with Relative Complement
 * $\complement_S \left({T}\right) \cup T = S$
 * 1) Set with Relative Complement forms Partition
 * Let $\varnothing \subsetneqq T \subsetneqq S$.
 * Then $\left\{{T, \complement_S \left({T}\right)}\right\}$ is a partition of $S$.
 * 1) Set Difference as Intersection with Relative Complement
 * Let $A, B \subseteq S$.
 * $A \setminus B = A \cap \relcomp S B$

Symmetric Difference

 * 1) Equivalence of Definitions of Symmetric Difference
 * $S * T := \paren {S \setminus T} \cup \paren {T \setminus S}$
 * $S * T = \paren {S \cup T} \setminus \paren {S \cap T}$
 * $S * T = \left({S \cap \overline T}\right) \cup \left({\overline S \cap T}\right)$
 * $S * T = \left({S \cup T}\right) \cap \left({\overline S \cup \overline T}\right)$
 * $S * T := \left\{{x: x \in S \oplus x \in T}\right\}$
 * 1) Symmetric Difference is Commutative
 * $S * T = T * S$
 * 1) Symmetric Difference with Self is Empty Set
 * $S * S = \O$
 * 1) Symmetric Difference of Equal Sets
 * $S = T \iff S * T = \O$
 * 1) Symmetric Difference with Empty Set
 * $S * \O = S$
 * 1) Intersection Distributes over Symmetric Difference
 * $\paren {R * S} \cap T = \paren {R \cap T} * \paren {S \cap T}$
 * $T \cap \paren {R * S} = \paren {T \cap R} * \paren {T \cap S}$
 * 1) Symmetric Difference of Unions
 * $\left({R \cup T}\right) * \left({S \cup T}\right) = \left({R * S}\right) \setminus T$
 * 1) Symmetric Difference is Associative
 * $R * \paren {S * T} = \paren {R * S} * T$

Universe

 * 1) Intersection with Universe
 * $\mathbb U \cap S = S$
 * 1) Union with Universe
 * $\mathbb U \cup S = \mathbb U$
 * 1) Complement of Empty Set is Universe
 * $\complement \paren \O = \mathbb U$
 * 1) Complement of Universe is Empty Set
 * $\complement \paren {\mathbb U} = \O$
 * 1) Complement of Complement
 * $\map \complement {\map \complement S} = S$
 * 1) Intersection with Complement
 * $S \cap \complement \left({S}\right) = \varnothing$
 * 1) Union with Complement
 * $S \cup \complement \left({S}\right) = \mathbb U$
 * 1) Set with Complement forms Partition
 * Let $\varnothing \subset S \subset \mathbb U$.
 * Then $S$ and its complement $\complement \left({S}\right)$ form a partition of the universal set $\mathbb U$.
 * 1) Set Difference as Intersection with Complement
 * $A \setminus B = A \cap \complement \left({B}\right)$
 * 1) Intersection with Complement is Empty iff Subset
 * $S \subseteq T \iff S \cap \complement \paren T = \O$
 * 1) Set Complement inverts Subsets
 * $S \subseteq T \iff \complement \left({T}\right) \subseteq \complement \left({S}\right)$
 * 1) Set Difference of Complements
 * $\complement \left({S}\right) \setminus \complement \left({T}\right) = T \setminus S$
 * 1) Empty Intersection iff Subset of Complement
 * $S \cap T = \varnothing \iff S \subseteq \complement \paren T$
 * 1) Symmetric Difference of Complements
 * $\map \complement S * \map \complement T = S * T$
 * 1) Symmetric Difference with Universe
 * $\mathbb U * S = \complement \paren S$
 * 1) De Morgan's Laws (Set Theory)/Set Complement/Complement of Union
 * $\overline {T_1 \cup T_2} = \overline T_1 \cap \overline T_2$
 * 1) De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection
 * $\overline {T_1 \cap T_2} = \overline T_1 \cup \overline T_2$
 * 1) Symmetric Difference with Complement
 * $S * \relcomp {} S = \mathbb U$

Cartesian Product

 * 1) Equality of Ordered Pairs
 * $\tuple {a, b} = \tuple {c, d} \iff a = c \land b = d$
 * 1) Cartesian Product is Empty iff Factor is Empty
 * $S \times T = \O \iff S = \O \lor T = \O$
 * 1) Cartesian Product is Anticommutative
 * Let $S, T \ne \O$.
 * Then $S \times T = T \times S \implies S = T$
 * 1) Cartesian Product of Subsets
 * Let $A, B, S, T$ be sets such that $A \subseteq B$ and $S \subseteq T$.
 * Then $A \times S \subseteq B \times T$.
 * Also $A \times S \subseteq B \times T \iff A \subseteq B \land S \subseteq T$.
 * 1) Cartesian Product of Intersections
 * $\paren {S_1 \cap S_2} \times \paren {T_1 \cap T_2} = \paren {S_1 \times T_1} \cap \paren {S_2 \times T_2}$
 * 1) Cartesian Product of Unions
 * $\paren {S_1 \cup S_2} \times \paren {T_1 \cup T_2} = \paren {S_1 \times T_1} \cup \paren {S_2 \times T_2} \cup \paren {S_1 \times T_2} \cup \paren {S_2 \times T_1}$
 * 1) Cartesian Product Distributes over Union
 * $A \times \paren {B \cup C} = \paren {A \times B} \cup \paren {A \times C}$
 * $\paren {B \cup C} \times A = \paren {B \times A} \cup \paren {C \times A}$
 * 1) Cartesian Product Distributes over Set Difference
 * $S \times \left({T_1 \setminus T_2}\right) = \left({S \times T_1}\right) \setminus \left({S \times T_2}\right)$
 * $\left({T_1 \setminus T_2}\right) \times S = \left({T_1 \times S}\right) \setminus \left({T_2 \times S}\right)$
 * 1) Set Difference of Cartesian Products
 * $\paren {S_1 \times S_2} \setminus \paren {T_1 \times T_2} = \paren {S_1 \times \paren {S_2 \setminus T_2} } \cup \paren {\paren {S_1 \setminus T_1} \times S_2}$