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

Images

 * 1) Image of Singleton under Relation
 * $\forall s \in S: \mathcal R \left({s}\right) = \mathcal R \left[{\left\{{s}\right\}}\right]$
 * 1) Image of Subset under Relation is Subset of Image
 * $A \subseteq B \implies \mathcal R \left[{A}\right] \subseteq \mathcal R \left[{B}\right]$
 * 1) Image of Element is Subset
 * $s \in A \implies \mathcal R \left({s}\right) \subseteq \mathcal R \left[{A}\right]$
 * 1) Image is Subset of Codomain
 * $\forall A \subseteq \operatorname{Dom} \left ({\mathcal R}\right): \mathcal R \left({A}\right) \subseteq T$
 * 1) Image of Empty Set is Empty Set
 * $\mathcal R \left[{\varnothing}\right] = \varnothing$

Composition
$\mathcal R$ is one-to-many. $\mathcal R$ is one-to-many.
 * 1) Domain of Composite Relation
 * $\operatorname{Dom} \left ({\mathcal R_2 \circ \mathcal R_1}\right) = \operatorname{Dom} \left ({\mathcal R_1}\right)$
 * 1) Composition of Relations is Associative
 * $\paren {\mathcal R_3 \circ \mathcal R_2} \circ \mathcal R_1 = \mathcal R_3 \circ \paren {\mathcal R_2 \circ \mathcal R_1}$
 * 1) Inverse of Inverse Relation
 * $\left({\mathcal R^{-1}}\right)^{-1} = \mathcal R$
 * 1) Preimage of Relation is Subset of Domain
 * $\operatorname{Im}^{-1} \left({\mathcal R}\right) \subseteq S$
 * 1) Inverse of Composite Relation
 * $\left({\mathcal R_2 \circ \mathcal R_1}\right)^{-1} = \mathcal R_1^{-1} \circ \mathcal R_2^{-1}$
 * 1) Image of Union under Relation
 * $\mathcal R \sqbrk {S_1 \cup S_2} = \mathcal R \sqbrk {S_1} \cup \mathcal R \sqbrk {S_2}$
 * 1) Image of Intersection under Relation
 * $\mathcal R \sqbrk {S_1 \cap S_2} \subseteq \mathcal R \sqbrk {S_1} \cap \mathcal R \sqbrk {S_2}$
 * 1) Image of Set Difference under Relation
 * $\mathcal R \sqbrk A \setminus \mathcal R \sqbrk B \subseteq \mathcal R \sqbrk {A \setminus B}$
 * 1) Preimage of Union under Relation
 * $\mathcal R^{-1} \left[{T_1 \cup T_2}\right] = \mathcal R^{-1} \left[{T_1}\right] \cup \mathcal R^{-1} \left[{T_2}\right]$
 * 1) Preimage of Intersection under Relation
 * $\mathcal R^{-1} \left[{C \cap D}\right] \subseteq \mathcal R^{-1} \left[{C}\right] \cap \mathcal R^{-1} \left[{D}\right]$
 * 1) Preimage of Set Difference under Relation
 * $\mathcal R^{-1} \left[{C}\right] \setminus \mathcal R^{-1} \left[{D}\right] \subseteq \mathcal R^{-1} \left[{C \setminus D}\right]$
 * 1) Image of Subset is Image of Restriction
 * Let $f: S \to T$ be a mapping, $X \subseteq S$, and $f \restriction_X$ be the restriction of $f$ to $X$.
 * Then $f \left[{X}\right] = \operatorname{Im} \left({f \restriction_X}\right)$
 * 1) Inverse of Many-to-One Relation is One-to-Many
 * The inverse of a many-to-one relation is a one-to-many relation, and vice versa.
 * 1) Image of Intersection under One-to-Many Relation
 * Let $S$ and $T$ be sets and $\mathcal R \subseteq S \times T$ be a relation.
 * Then $\forall S_1, S_2 \subseteq S: \mathcal R \left[{S_1 \cap S_2}\right] = \mathcal R \left[{S_1}\right] \cap \mathcal R \left[{S_2}\right]$
 * 1) One-to-Many Image of Set Difference
 * Let $\mathcal R \subseteq S \times T$ be a relation and $A$ and $B$ be subsets of $S$.
 * Then $\quad \mathcal R \left[{A}\right] \setminus \mathcal R \left[{B}\right] = \mathcal R \left[{A \setminus B}\right]$

Properties

 * 1) Equivalence of Definitions of Reflexive Relation
 * $\mathcal R$ is reflexive $\forall x \in S: \tuple {x, x} \in \mathcal R$
 * $\mathcal R$ is reflexive it is a superset of the diagonal relation: $\Delta_S \subseteq \mathcal R$
 * 1) Equivalence of Definitions of Symmetric Relation
 * $\tuple {x, y} \in \mathcal R \implies \tuple {y, x} \in \mathcal R$
 * $\mathcal R^{-1} = \mathcal R$
 * $\mathcal R \subseteq \mathcal R^{-1}$
 * 1) Equivalence of Definitions of Transitive Relation
 * $\mathcal R$ is transitive $\tuple {x, y} \in \mathcal R \land \tuple {y, z} \in \mathcal R \implies \tuple {x, z} \in \mathcal R$.
 * $\mathcal R$ is transitive $\mathcal R \circ \mathcal R \subseteq \mathcal R$.
 * 1) Relation Reflexivity
 * Every relation has exactly one of these properties: it is either: reflexive, antireflexive or non-reflexive.
 * 1) Relation Symmetry
 * Every non-null relation has exactly one of these properties: it is either symmetric, asymmetric or non-symmetric.
 * 1) Relation is Reflexive Symmetric and Antisymmetric iff Diagonal Relation
 * $\mathcal R$ is reflexive, symmetric and antisymmetric $\mathcal R$ is the diagonal relation $\Delta_S$.
 * 1) Relation both Symmetric and Asymmetric is Null
 * Let $\mathcal R$ be a relation in $S$ which is both symmetric and asymmetric.
 * Then $\mathcal R = \varnothing$
 * 1) Relation is Symmetric and Antisymmetric iff Coreflexive
 * 2) Asymmetric Relation is Antisymmetric
 * 3) Asymmetric Relation is Antireflexive
 * 4) Antireflexive and Transitive Relation is Asymmetric
 * 5) Antireflexive and Transitive Relation is Antisymmetric
 * 6) Antitransitive Relation is Antireflexive
 * 7) Symmetric Transitive and Serial Relation is Reflexive
 * Let $\mathcal R$ be a relation which is symmetric, transitive, and serial.
 * Then $\mathcal R$ is reflexive. Thus such a relation is an equivalence.

Inverse Relations

 * 1) Inverse of Reflexive Relation is Reflexive
 * Let $\mathcal R$ be a relation on a set $S$.
 * If $\mathcal R$ is reflexive, then so is $\mathcal R^{-1}$.
 * 1) Inverse of Antireflexive Relation is Antireflexive
 * 2) Inverse of Non-Reflexive Relation is Non-Reflexive
 * 3) Inverse of Symmetric Relation is Symmetric
 * 4) Inverse of Antisymmetric Relation is Antisymmetric
 * 5) Inverse of Asymmetric Relation is Asymmetric
 * 6) Inverse of Non-Symmetric Relation is Non-Symmetric
 * 7) Inverse of Transitive Relation is Transitive
 * 8) Inverse of Antitransitive Relation is Antitransitive
 * 9) Inverse of Non-Transitive Relation is Non-Transitive

Restriction of Relation

 * 1) Restriction of Reflexive Relation is Reflexive
 * 2) Restriction of Antireflexive Relation is Antireflexive
 * 3) Restriction of Symmetric Relation is Symmetric
 * 4) Restriction of Antisymmetric Relation is Antisymmetric
 * 5) Restriction of Asymmetric Relation is Asymmetric
 * 6) Restriction of Transitive Relation is Transitive
 * 7) Restriction of Antitransitive Relation is Antitransitive
 * 8) Restriction of Connected Relation is Connected
 * 9) Restriction of Serial Relation is Not Necessarily Serial
 * 10) Restriction of Non-Reflexive Relation is Not Necessarily Non-Reflexive
 * 11) Restriction of Non-Symmetric Relation is Not Necessarily Non-Symmetric
 * 12) Restriction of Non-Transitive Relation is Not Necessarily Non-Transitive
 * 13) Restriction of Non-Connected Relation is Not Necessarily Non-Connected


 * 1) Inverse Relation Equal iff Subset
 * If a relation $\mathcal R$ is a subset or superset of its inverse, then it equals its inverse.
 * 1) Composition of Relation with Inverse is Symmetric
 * Let $\mathcal R \subseteq S \times T$ be a relation.
 * Then the composition of $\mathcal R$ with its inverse $\mathcal R^{-1}$ is symmetric.
 * 1) Many-to-One Relation Composite with Inverse is Transitive
 * Let $\mathcal R \subseteq S \times T$ be a relation which is many-to-one.
 * Then the composites (both ways) of $\mathcal R$ and its inverse $\mathcal R^{-1}$, that is, both $\mathcal R^{-1} \circ \mathcal R$ and $\mathcal R \circ \mathcal R^{-1}$, are transitive.

Equivalence Relations

 * 1) Equivalence of Definitions of Equivalence Relation
 * $\mathcal R$ is an equivalence relation it is reflexive, symmetric, and transitive.
 * $\mathcal R$ is an equivalence relation : $\Delta_S \cup \mathcal R^{-1} \cup \mathcal R \circ \mathcal R \subseteq \mathcal R$
 * 1) Diagonal Relation is Equivalence
 * The diagonal relation $\Delta_S$ on a set $S$ is always an equivalence in $S$.
 * 1) Trivial Relation is Equivalence
 * The trivial relation on $S$: $\mathcal R = S \times S$ is always an equivalence in $S$.
 * 1) Element in its own Equivalence Class
 * Let $\mathcal R$ be an equivalence relation on a set $S$.
 * Then every element of $S$ is in its own $\mathcal R$-class: $\forall x \in S: x \in \eqclass x {\mathcal R}$
 * 1) Equivalence Class of Element is Subset
 * Let $\mathcal R$ be an equivalence relation on a set $S$.
 * The $\mathcal R$-class of every element of $S$ is a subset of the set the element is in: $\forall x \in S: \left[\!\left[{x}\right]\!\right]_\mathcal R \subseteq S$
 * 1) Equivalence Class is not Empty
 * Let $\mathcal R$ be an equivalence relation on a set $S$.
 * Then no $\mathcal R$-class is empty.
 * 1) Equivalence Class holds Equivalent Elements
 * Let $\mathcal R$ be an equivalence relation on a set $S$.
 * Then: $\tuple {x, y} \in \mathcal R \iff \eqclass x {\mathcal R} = \eqclass y {\mathcal R}$
 * 1) Equivalence Classes are Disjoint
 * Let $\mathcal R$ be an equivalence relation on a set $S$.
 * Then all $\mathcal R$-classes are pairwise disjoint: $\tuple {x, y} \notin \mathcal R \iff \eqclass x {\mathcal R} \cap \eqclass y {\mathcal R} = \O$
 * 1) Fundamental Theorem on Equivalence Relations
 * Let $\mathcal R \subseteq S \times S$ be an equivalence on a set $S$.
 * Then the quotient $S / \mathcal R$ of $S$ by $\mathcal R$ forms a partition of $S$.
 * 1) Equivalence Class is Unique
 * Let $\mathcal R$ be an equivalence relation on $S$.
 * For each $x \in S$, the one and only one $\mathcal R$-class to which $x$ belongs is $\eqclass x {\mathcal R}$.
 * 1) Union of Equivalence Classes is Whole Set
 * Let $\mathcal R \subseteq S \times S$ be an equivalence on a set $S$.
 * Then the set of $\mathcal R$-classes constitutes the whole of $S$.
 * 1) Intersection of Equivalences
 * The intersection of two equivalence relations is itself an equivalence relation.
 * 1) Union of Equivalences
 * The union of two equivalence relations is not necessarily an equivalence relation itself.
 * 1) Equivalence iff Diagonal and Inverse Composite
 * Let $\mathcal R$ be a relation on $S$.
 * Then $\mathcal R$ is an equivalence relation on $S$ iff $\Delta_S \subseteq \mathcal R$ and $\mathcal R = \mathcal R \circ \mathcal R^{-1}$.