design/tpl: Mostly-complex symbol index entries for Chapter 0
parent
7624bd2958
commit
7f4fc8e3b7
|
|
@ -20,11 +20,7 @@ When you see any of these prefixed with ``0.'',
|
|||
|
||||
|
||||
\subsection{Propositional Logic}
|
||||
\indexsym{\true,\false}{boolean}
|
||||
\index{boolean!FALSE@\tamefalse{} (\false)}%
|
||||
\index{boolean!TRUE@\tametrue{} (\true)}%
|
||||
\indexsym\Int{integer}
|
||||
\index{integer (\Int)}%
|
||||
\index{logic!propositional}
|
||||
We reproduce here certain axioms and corollaries of propositional logic for
|
||||
convenience and to clarify our interpretation of certain concepts.
|
||||
The use of the symbols $\logand$, $\logor$, and~$\neg$ are standard.
|
||||
|
|
@ -49,14 +45,19 @@ We further use $\equiv$ in place of $\leftrightarrow$ to represent material
|
|||
$p \infer (p\logor q)$ and $q \infer (p\logor q)$.
|
||||
\end{definition}
|
||||
|
||||
\indexsym\neg{negation}
|
||||
\index{negation (\ensuremath{\neg})}
|
||||
\index{law of excluded middle}
|
||||
\begin{definition}[Law of Excluded Middle]
|
||||
$\infer (p \logor \neg p)$.
|
||||
\end{definition}
|
||||
|
||||
\index{law of non-contradiction}
|
||||
\begin{definition}[Law of Non-Contradiction]
|
||||
$\infer \neg(p \logand \neg p)$.
|
||||
\end{definition}
|
||||
|
||||
\index{De Morgan's theorem}
|
||||
\begin{definition}[De Morgan's Theorem]
|
||||
$\neg(p \logand q) \infer (\neg p \logor \neg q)$
|
||||
and $\neg(p \logor q) \infer (\neg p \logand \neg q)$.
|
||||
|
|
@ -72,59 +73,69 @@ $\equiv$ denotes a logical identity.
|
|||
Consequently,
|
||||
it'll often be used as a definition operator.
|
||||
|
||||
\begin{definition}[Logical Implication]
|
||||
\indexsym{\!\!\implies\!\!}{implication}
|
||||
\index{implication (\ensuremath{\implies})}
|
||||
\begin{definition}[Implication]
|
||||
$p\implies q \infer (\neg p \logor q)$.
|
||||
\end{definition}
|
||||
|
||||
\indexsym{\true}{boolean, true}
|
||||
\indexsym{\false}{boolean, false}
|
||||
\index{boolean!FALSE@\tamefalse{} (\false)}%
|
||||
\index{boolean!TRUE@\tametrue{} (\true)}%
|
||||
\begin{definition}[Truth Values]\dfnlabel{truth-values}
|
||||
$\infer\true$ and $\infer\neg\false$.
|
||||
\end{definition}
|
||||
|
||||
|
||||
\subsection{First-Order Logic and Set Theory}
|
||||
\index{first-order logic}
|
||||
\index{logic!first-order}
|
||||
\indexsym\emptyset{set empty}
|
||||
\indexsym{\Set{}}{set}
|
||||
\index{set!empty (\ensuremath{\emptyset, \{\}})}
|
||||
The symbol $\emptyset$ represents the empty set---%
|
||||
the set of zero elements.
|
||||
We assume that the axioms of ZFC~set theory hold,
|
||||
but define $\in$ here for clarity.
|
||||
|
||||
% TODO: set-builder notation, union, intersection
|
||||
\indexsym\in{set membership}
|
||||
\index{set!membership@membership (\ensuremath{\in})}
|
||||
\indexsym\union{set, union}
|
||||
\indexsym\intersect{set, intersection}
|
||||
\index{set!membership@membership (\ensuremath\in)}
|
||||
\index{set!union (\ensuremath\union)}
|
||||
\index{set!intersection (\ensuremath\intersect)}
|
||||
\begin{definition}[Set Membership]
|
||||
$x \in S \equiv \Set{x} \cap S \not= \emptyset.$
|
||||
$x \in S \equiv \Set{x} \intersect S \not= \emptyset.$
|
||||
\end{definition}
|
||||
|
||||
\indexsym\forall{quantification, universal}
|
||||
\indexsym\exists{quantification, existential}
|
||||
\index{quantification!universal (\ensuremath{\forall})}
|
||||
\index{quantification!existential (\ensuremath{\exists})}
|
||||
$\forall$ denotes first-order universal quantification (``for all''),
|
||||
and $\exists$ first-order existential quantification (``there exists''),
|
||||
over some domain.
|
||||
|
||||
\indexsym\logor{disjunction}
|
||||
\index{disjunction (\ensuremath{\logor})}
|
||||
\indexsym\exists{quantification, existential}
|
||||
\index{quantification!existential (\ensuremath\exists)}
|
||||
\begin{definition}[Existential Quantification]\dfnlabel{exists}
|
||||
$\Exists{x\in X}{P(x)} \equiv
|
||||
\true \in \Set{P(x) \mid x\in X}$.
|
||||
\end{definition}
|
||||
|
||||
\indexsym\logand{conjunction}
|
||||
\index{conjunction (\ensuremath{\logand})}
|
||||
\indexsym\forall{quantification, universal}
|
||||
\index{quantification!universal (\ensuremath\forall)}
|
||||
\begin{definition}[Universal Quantification]\dfnlabel{forall}
|
||||
$\Forall{x\in X}{P(x)} \equiv \neg\Exists{x\in X}{\neg P(x)}$.
|
||||
\end{definition}
|
||||
|
||||
\indexsym\emptyset{set empty}
|
||||
\indexsym{\Set{}}{set}
|
||||
\index{set!empty (\ensuremath{\emptyset, \{\}})}
|
||||
\index{quantification!vacuous truth}
|
||||
\begin{remark}[Vacuous Truth]
|
||||
By Definition~7, $\Exists{x\in\emptyset}P \equiv \false$
|
||||
By \dfnref{exists}, $\Exists{x\in\emptyset}P \equiv \false$
|
||||
and by \dfnref{forall}, $\Forall{x\in\emptyset}P \equiv \true$.
|
||||
And so we also have the tautologies $\infer \neg\Exists{x\in\emptyset}P$
|
||||
and $\infer \Forall{x\in\emptyset}P$.
|
||||
\end{remark}
|
||||
|
||||
\indexsym\Int{integer}
|
||||
\index{integer (\Int)}%
|
||||
\begin{definition}[Boolean/Integer Equivalency]\dfnlabel{bool-int}
|
||||
$\Set{0,1}\in\Int, \false \equiv 0$ and $\true \equiv 1$.
|
||||
\end{definition}
|
||||
|
|
@ -137,10 +148,22 @@ $\forall$ denotes first-order universal quantification (``for all''),
|
|||
|
||||
|
||||
\subsection{Functions}
|
||||
\indexsym{f, g}{function}
|
||||
\indexsym\mapsto{function, map}
|
||||
\indexsym\rightarrow{function, domain map}
|
||||
\index{function}
|
||||
\index{function!map (\ensuremath\mapsto)}
|
||||
\index{map|see {function}}
|
||||
\index{function!domain}
|
||||
\index{function!codomain}
|
||||
\index{domain|see {function, domain}}
|
||||
\index{function!domain map (\ensuremath\rightarrow)}
|
||||
The notation $f = x \mapsto x' : A\rightarrow B$ represents a function~$f$
|
||||
that maps from~$x$ to~$x'$,
|
||||
where $x\in A$ (the domain of~$f$) and $x'\in B$ (the co-domain of~$f$).
|
||||
|
||||
\indexsym\times{set, Cartesian product}
|
||||
\index{set!Cartesian product (\ensuremath\times)}
|
||||
A function $A\rightarrow B$ can be represented as the Cartesian
|
||||
product of its domain and codomain, $A\times B$.
|
||||
For example,
|
||||
|
|
@ -151,6 +174,9 @@ For example,
|
|||
\Set{\ldots,\,(0,0),\,(1,1),\,(2,4),\,(3,9),\,\ldots}.
|
||||
\end{equation*}
|
||||
|
||||
\indexsym{[\,]}{function, image}
|
||||
\index{function!image (\ensuremath{[\,]})}
|
||||
\index{function!as a set}
|
||||
The set of values over which some function~$f$ ranges is its \emph{image},
|
||||
which is a subset of its codomain.
|
||||
In the example above,
|
||||
|
|
@ -170,6 +196,9 @@ We therefore have
|
|||
f[] &= f[A].
|
||||
\end{align}
|
||||
|
||||
\indexsym{()}{tuple}
|
||||
\index{tuple (\ensuremath{()})}
|
||||
\index{relation|see {function}}
|
||||
And ordered pair $(x,y)$ is also called a \emph{$2$-tuple}.
|
||||
Generally,
|
||||
an \emph{$n$-tuple} is used to represent an $n$-ary function,
|
||||
|
|
@ -190,13 +219,13 @@ Binary functions are often written using \emph{infix} notation;
|
|||
|
||||
|
||||
\subsubsection{Binary Operations On Functions}
|
||||
\indexsym{R}{relation}
|
||||
Consider two unary functions $f$ and~$g$,
|
||||
and a binary relation~$R$.
|
||||
\indexsym{\bicomp{R}}{function, binary composition}
|
||||
\index{function!binary composition (\ensuremath{\bicomp{R}})}
|
||||
We introduce a notation~$\bicomp R$ to denote the composition of a binary
|
||||
function with two unary functions.\footnote{%
|
||||
The notation originates from~$\circ$ to denote ordinary function
|
||||
composition,
|
||||
as in $(f\circ g)(x) = f(g(x))$.}
|
||||
function with two unary functions.
|
||||
|
||||
\begin{align}
|
||||
f &: A \rightarrow B \\
|
||||
|
|
@ -205,11 +234,14 @@ We introduce a notation~$\bicomp R$ to denote the composition of a binary
|
|||
f \bicomp{R} g &= \alpha \mapsto f_\alpha R g_\alpha : A \rightarrow F
|
||||
\end{align}
|
||||
|
||||
\indexsym\circ{function, composition}
|
||||
\index{function!composition (\ensuremath\circ)}
|
||||
Note that $f$ and~$g$ must share the same domain~$A$.
|
||||
In that sense,
|
||||
this is the mapping of the operation~$R$ over the domain~$A$.
|
||||
This is analogous to unary function composition~$f\circ g$.
|
||||
|
||||
\index{function!constant}
|
||||
A scalar value~$x$ can be mapped onto some function~$f$ using a constant
|
||||
function.
|
||||
For example,
|
||||
|
|
@ -219,8 +251,12 @@ For example,
|
|||
f \bicomp+ (\_\mapsto x) = \alpha \mapsto f_\alpha + x.
|
||||
\end{equation*}
|
||||
|
||||
\indexsym{\_}{variable, wildcard}
|
||||
\index{variable!wildcard/hole (\ensuremath{\_})}
|
||||
The symbol~$\_$ is used to denote a variable that is never referenced.
|
||||
|
||||
\indexsym{\bicompi{R}}{function, binary composition, recursive}
|
||||
\index{function!binary composition (\ensuremath{\bicomp{R}})!recursive (\ensuremath{\bicompi{R}})}
|
||||
For convenience,
|
||||
we also define $\bicompi{R}$,
|
||||
which recursively handles combinations of function and scalar values.
|
||||
|
|
@ -255,6 +291,16 @@ Unfortunately,
|
|||
A vector is a sequence of values, defined as a function of
|
||||
an index~set.
|
||||
|
||||
% TODO: font changes in index, making langle unavailable
|
||||
%\indexsym{\Vector{}}{vector}
|
||||
\index{vector!definition (\ensuremath{\Vector{}})}
|
||||
\index{sequence|see vector}
|
||||
\indexsym\Vectors{vector}
|
||||
\index{real number (\ensuremath\Real)}
|
||||
\indexsym\Real{real number}
|
||||
\indexsym{\Fam{a}jJ}{index set}
|
||||
\index{family|see {index set}}
|
||||
\index{index set (\ensuremath{\Fam{a}jJ})}
|
||||
\begin{definition}[Vector]\dfnlabel{vec}
|
||||
Let $J\subset\Int$ represent an index set.
|
||||
A \emph{vector}~$v\in\Vectors^\Real$ is a totally ordered sequence of
|
||||
|
|
@ -270,6 +316,7 @@ This definition means that $v_j = v(j)$,
|
|||
We may omit the superscript such that $\Vectors^\Real=\Vectors$
|
||||
and $\Vector{\ldots}^\Real=\Vector{\ldots}$.
|
||||
|
||||
\index{vector!matrix}
|
||||
\begin{definition}[Matrix]\dfnlabel{matrix}
|
||||
Let $J\subset\Int$ represent an index set.
|
||||
A \emph{matrix}~$M\in\Matrices$ is a totally ordered sequence of
|
||||
|
|
@ -318,7 +365,8 @@ We can also add two vectors, and scale them:
|
|||
|
||||
|
||||
\subsection{XML Notation}
|
||||
\index{XML}
|
||||
\indexsym{\xml{<>}}{XML}
|
||||
\index{XML!notation (\xml{<>})}
|
||||
The grammar of \tame{} is XML.
|
||||
Equivalence relations will be used to map source expressions to an
|
||||
underlying mathematical expression.
|
||||
|
|
|
|||
|
|
@ -77,6 +77,7 @@
|
|||
\newcommand\Matrices{\ensuremath{\Vectors^{\Vectors^\Real}}}
|
||||
|
||||
\let\union\cup
|
||||
\let\intersect\cap
|
||||
\let\infer\vdash
|
||||
|
||||
\newcommand\len[1]{\ensuremath{\left|#1\right|}}
|
||||
|
|
|
|||
Loading…
Reference in New Issue