9.11: Summary of Relational Properties
- Page ID
- 48354
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A relation \(R : A \rightarrow A\) is the same as a digraph with vertices \(A\).
Reflexivity \(R\) is reflexive when
\[\nonumber \forall x \in A. x R x.\]
Every vertex in \(R\) has a self-loop.
Irreflexivity \(R\) is irreflexive when
\[\nonumber \text{NOT}[\exists x \in A. x R x].\]
There are no self-loops in \(R\).
Symmetry \(R\) is symmetric when
\[\nonumber \forall x,y \in A. x R y { IMPLIES } y R x.\]
If there is an edge from \(x\) to \(y\) in \(R\), then there is an edge back from \(y\) to \(x\) as well.
Asymmetry \(R\) is asymmetric when
\[\nonumber \forall x,y \in A. x R y { IMPLIES NOT } y R x.\]
There is at most one directed edge between any two vertices in \(R\), and there are no self-loops.
Antisymmetry R is antisymmetric when
\[\nonumber \forall x \neq y \in A. x R y { IMPLIES NOT } y R x.\]
Equivalently,
\[\nonumber \forall x,y \in A. (x R y { AND } y R x) \text{ IMPLIES } x = y.\]
There is at most one directed edge between any two distinct vertices, but there may be self-loops.
Transitivity \(R\) is transitive when
\[\nonumber \forall x,y,z \in A. (x R y { AND } y R z) \text{ IMPLIES } x R z.\]
If there is a positive length path from \(u\) to \(v\), then there is an edge from \(u\) to \(v\).
Linear \(R\) is linear when
\[\nonumber \forall x \neq y \in A. (x R y { OR } y R x)\]
Given any two vertices in \(R\), there is an edge in one direction or the other between them.
For any finite, nonempty set of vertices of \(R\), there is a directed path going through exactly these vertices.
Strict Partial Order \(R\) is a strict partial order iff \(R\) is transitive and irreflexive iff \(R\) is transitive and asymmetric iff it is the positive length walk relation of a DAG.
Weak Partial Order \(R\) is a weak partial order iff \(R\) is transitive and anti-symmetric and reflexive iff \(R\) is the walk relation of a DAG.
Equivalence Relation \(R\) is an equivalence relation iff \(R\) is reflexive, symmetric and transitive iff \(R\) equals the in-the-same-block-relation for some partition of domain(\(R\)).