Skip to main content ☰ Contents Index You! < Prev ^ Up Next > \(\require{cancel}
\newcommand{\nth}[1][n]{{#1}^{\mathrm{th}}}
\newcommand{\bbrac}[1]{\bigl(#1\bigr)}
\newcommand{\Bbrac}[1]{\Bigl(#1\Bigr)}
\newcommand{\correct}{\boldsymbol{\checkmark}}
\newcommand{\incorrect}{\boldsymbol{\times}}
\newcommand{\inv}[2][1]{{#2}^{-{#1}}}
\newcommand{\leftsub}[3][1]{\mathord{{}_{#2\mkern-#1mu}#3}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\I}{\mathbb{I}}
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}
\DeclareMathOperator{\sqrtop}{sqrt}
\newcommand{\lgcnot}{\neg}
\newcommand{\lgcand}{\wedge}
\newcommand{\lgcor}{\vee}
\newcommand{\lgccond}{\rightarrow}
\newcommand{\lgcbicond}{\leftrightarrow}
\newcommand{\lgcimplies}{\Rightarrow}
\newcommand{\lgcequiv}{\Leftrightarrow}
\newcommand{\lgctrue}{\mathrm{T}}
\newcommand{\lgcfalse}{\mathrm{F}}
\newcommand{\boolnot}[1]{{#1}'}
\newcommand{\boolzero}{\mathbf{0}}
\newcommand{\boolone}{\mathbf{1}}
\newcommand{\setdef}[2]{\left\{\mathrel{}#1\mathrel{}\middle|\mathrel{}#2\mathrel{}\right\}}
\newcommand{\inlinesetdef}[2]{\{\mathrel{}#1\mathrel{}\mid\mathrel{}#2\mathrel{}\}}
\let\emptyword\emptyset
\renewcommand{\emptyset}{\varnothing}
\newcommand{\relcmplmnt}{\smallsetminus}
\newcommand{\union}{\cup}
\newcommand{\intersection}{\cap}
\newcommand{\cmplmnt}[1]{{#1}^{\mathrm{c}}}
\newcommand{\disjunion}{\sqcup}
\newcommand{\cartprod}{\times}
\newcommand{\words}[1]{{#1}^\ast}
\newcommand{\length}[1]{\abs{#1}}
\newcommand{\powsetbare}{\mathcal{P}}
\newcommand{\powset}[1]{\powsetbare(#1)}
\newcommand{\funcdef}[4][\to]{#2\colon #3 #1 #4}
\newcommand{\ifuncto}{\hookrightarrow}
\newcommand{\ifuncdef}[3]{\funcdef[\ifuncto]{#1}{#2}{#3}}
\newcommand{\sfuncto}{\twoheadrightarrow}
\newcommand{\sfuncdef}[3]{\funcdef[\sfuncto]{#1}{#2}{#3}}
\newcommand{\funcgraphbare}{\Delta}
\newcommand{\funcgraph}[1]{\funcgraphbare(#1)}
\newcommand{\relset}[3]{#1_{{} #2 #3}}
\newcommand{\gtset}[2]{\relset{#1}{\gt}{#2}}
\newcommand{\posset}[1]{\gtset{#1}{0}}
\newcommand{\geset}[2]{\relset{#1}{\ge}{#2}}
\newcommand{\nnegset}[1]{\geset{#1}{0}}
\newcommand{\neqset}[2]{\relset{#1}{\neq}{#2}}
\newcommand{\nzeroset}[1]{\neqset{#1}{0}}
\newcommand{\ltset}[2]{\relset{#1}{\lt}{#2}}
\newcommand{\leset}[2]{\relset{#1}{\le}{#2}}
\newcommand{\natnumlt}[1]{\ltset{\N}{#1}}
\DeclareMathOperator{\id}{id}
\newcommand{\inclfunc}[2]{\iota_{#1}^{#2}}
\newcommand{\projfunc}[1]{\rho_{#1}}
\DeclareMathOperator{\proj}{proj}
\newcommand{\funcres}[2]{\left.{#1}\right\rvert_{#2}}
\newcommand{\altfuncres}[2]{\left.{#1}\right\rvert{#2}}
\DeclareMathOperator{\res}{res}
\DeclareMathOperator{\flr}{flr}
\newcommand{\floor}[1]{\lfloor {#1} \rfloor}
\newcommand{\funccomp}{\circ}
\newcommand{\funcinvimg}[2]{\inv{#1}\left({#2}\right)}
\newcommand{\card}[1]{\left\lvert #1 \right\rvert}
\DeclareMathOperator{\cardop}{card}
\DeclareMathOperator{\ncardop}{\#}
\newcommand{\EngAlphabet}{\{ \mathrm{a}, \, \mathrm{b}, \, \mathrm{c}, \, \dotsc, \, \mathrm{y}, \, \mathrm{z} \}}
\newcommand{\ShortEngAlphabet}{\{ \mathrm{a}, \, \mathrm{b}, \, \dotsc, \, \mathrm{z} \}}
\newcommand{\eqclass}[1]{\left[#1\right]}
\newcommand{\partorder}{\preceq}
\newcommand{\partorderstrict}{\prec}
\newcommand{\npartorder}{\npreceq}
\newcommand{\subgraph}{\preceq}
\newcommand{\subgraphset}[1]{\mathcal{S}(#1)}
\newcommand{\connectedsubgraphset}[1]{\mathcal{C}(#1)}
\newcommand{\permcomb}[3]{{#1}(#2,#3)}
\newcommand{\permcombalt}[3]{{#1}^{#2}_{#3}}
\newcommand{\permcombaltalt}[3]{{\leftsub{#2}{#1}}_{#3}}
\newcommand{\permutation}[2]{\permcomb{P}{#1}{#2}}
\newcommand{\permutationalt}[2]{\permcombalt{P}{#1}{#2}}
\newcommand{\permutationaltalt}[2]{{\permcombaltalt{P}{#1}{#2}}}
\newcommand{\combination}[2]{\permcomb{C}{#1}{#2}}
\newcommand{\combinationalt}[2]{\permcombalt{C}{#1}{#2}}
\newcommand{\combinationaltalt}[2]{{\permcombaltalt{C}{#1}{#2}}}
\newcommand{\choosefuncformula}[3]{\frac{#1 !}{#2 ! \, #3 !}}
\DeclareMathOperator{\matrixring}{M}
\newcommand{\uvec}[1]{\mathbf{#1}}
\newcommand{\zerovec}{\uvec{0}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 18.2 Basics and examples
What properties should a relation on a set have to be useful as a notion of “equivalence”?
Each object in the set should be equivalent to itself. So the relation should be
reflexive .
Equivalence should be bidirectional. That is, a pair of equivalent objects should be equivalent to
each other . So the relation should be
symmetric .
We should be able to infer equivalence from chains of equivalence. So the relation should be
transitive .
equivalence relation
\(\mathord{\equiv}\) symbol for an abstract equivalence relation (instead of the letter \(R\) that we’ve been using for abstract relations up until now)
Worked Example 18.2.1 .
Let
\(\mathscr{L}\) be the set of all possible logical statements built out of the statement variables
\(p_1, p_2, p_3, \dotsc\text{.}\) Show that
logical equivalence of statements is an equivalence relation on
\(\mathscr{L}\text{.}\)
Solution .
Reflexive.
We have \(A \lgcequiv A\) for every statement \(A\text{,}\) since \(A\) has the same truth table as itself.
Symmetric.
If \(A \lgcequiv B\text{,}\) then \(A,B\) have the same truth table, so \(B \lgcequiv A\text{.}\)
Transitive.
If \(A \lgcequiv B\) and \(B \lgcequiv C\text{,}\) then \(A\) has the same truth table as \(B\text{,}\) which has the same truth table as \(C\text{.}\) So \(A\) has the same truth table as \(C\text{,}\) i.e. \(A \lgcequiv C\text{.}\)
Here is an important equivalence relation on \(\N\) or on \(\Z\text{.}\)
equivalence modulo \(n\)
an equivalence of integers, where two integers are equivalent if they have the same remainder when divided by \(n\)
\(m_1 \equiv_n m_2\) integers \(m_1,m_2\) are equivalent modulo \(n\)
Checkpoint 18.2.2 .
Verify that equivalence modulo \(n\) is an equivalence relation.
