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

Relations

 * 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
 * $\mathcal R \left[{\varnothing}\right] = \varnothing$
 * 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
 * $\mathcal R^{-1} \left[{C}\right] \setminus \mathcal R^{-1} \left[{D}\right] \subseteq \mathcal R^{-1} \left[{C \setminus D}\right]$

Properties

 * 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

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