[translation] chapter 1 boolean expressions
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\end{lstlisting}
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\section{Boolean Operations}
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Boolesche Ausdr\"ucke sind Anweisungen, die zu \codeterm{wahr} oder
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\codeterm{falsch} ausgewertet werden. Man kennt sie z.B. aus der
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Mengenlehre. In der Programmierung werden sie eingesetzt, um z.B. die
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Beziehung zwischen Entit\"aten zu testen. Hierzu werden die
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\codeterm{relationalen Operatoren} (\code[Operator!relationaler!>]{>},
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\code[Operator!relationaler!<]{<},
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\code[Operator!relationaler!==]{==},
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\code[Operator!relationaler!"!]{!}, gr\"o{\ss}er als, kleiner als,
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gleich und nicht) eingesetzt. Mehrere Ausdr\"ucke werden mittels der
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\codeterm[Operator!logischer]{logischen Operatoren}
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(\code[Operator!logischer!and1@\&]{\&}, \code[Operator!logischer!or1@{"|} {}]{|},
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UND, ODER) verkn\"upft. Sie sind nicht nur wichtig, um
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Codeabschnitte bedingt auszuf\"uhren (Verzweigungen,
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\ref{controlstructsec}) sondern auch um aus Vektoren und Matrizen
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bequem Elemente auszuw\"ahlen (logisches Indizieren,
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\ref{logicalindexingsec}).
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Die Tabellen \ref{logicalandor} zeigen die Wahrheitstabellen f\"ur das
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logische UND, das logische ODER und das logische XOR
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(entweder-oder). Es werden die Aussagen A und B mit dem Operator
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verkn\"upft. Beim logischen UND ist der gesamte Ausdruck nur dann
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wahr, wenn beide Ausdr\"ucke sich zu wahr auswerten lassen. Anders
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ist das beim logischen ODER. Hier ist der gesamte Ausdruck wahr, wenn
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sich der eine \emph{oder} der andere Ausdruck, oder beide Ausdr\"ucke
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zu wahr auswerten lassen. Das auschlie{\ss}ende ODER (XOR) ist nur
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wahr, wenn entweder der eine oder der andere Ausdruck wahr ist und ist
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in \matlab{} als Funktion \code[xor()]{xor(A, B)} verf\"ugbar.
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\section{Boolean expressions}
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Boolean expressions are instructions that can be evaluated to
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\varcode{true} or \varcode{false}. In the context of programming they
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are used to test the relations accordingly the programming language
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defines operators for such instructions. The following
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\codeterm{relational operators} are defined:
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(\code[Operator!relational!>]{>}, \code[Operator!relational!<]{<},
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\code[Operator!relational!==]{==}, \code[Operator!relational!"~]{~},
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greater than, less than, equal to, and not. Via so called
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\codeterm[Operator!logical]{logical operators} it is possible to join
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single Boolean expressions (\code[Operator!logical!and1@\&]{\&},
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\code[Operator!logical!or1@{"|} {}]{|}, AND, OR). These expressions
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are important to control which parts of the code should be evaluated
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under a certain condition (conditional statements,
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Section~\ref{controlstructsec}) but also for accessing only certain
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elements of a vector or matrix (logical indexing,
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Section~\ref{logicalindexingsec}).
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The truth tables (\ref{logicalandor}) are used to visualize the
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results of Boolean expressions. The statements A and B can be
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evaluated to True or False. When they are combined with a logical AND
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the expression is true only if both statements are true. The logical
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OR, on the other hand, requires that at least one of the statements is
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true. The exclusive OR (XOR) is true only for cases in which one of
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the statements but not both are true. There is no operator for XOR in
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\matlab{} it is realized via the function \code[xor()]{xor(A,
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B)}.
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\begin{table}[tp]
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\titlecaption{Wahrheitstabellen logisches UND, ODER und XOR.}{}\label{logicalandor}
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\titlecaption{Truth tables for logical AND, OR and XOR.}{}\label{logicalandor}
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\begin{tabular}{llll}
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\multicolumn{2}{l}{\multirow{2}{*}{}} & \multicolumn{2}{c}{\textbf{B}} \\
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& \sffamily{\textbf{und}} & \multicolumn{1}{|c}{wahr} & falsch \\ \cline{2-4}
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\multirow{2}{*}{\textbf{A}} & \multicolumn{1}{l|}{wahr} & \multicolumn{1}{c}{\textcolor{mygreen}{wahr}} & \textcolor{red}{falsch} \erb \\
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& \multicolumn{1}{l|}{falsch} & \multicolumn{1}{l}{\textcolor{red}{falsch}} & \textcolor{red}{falsch}
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& \sffamily{\textbf{und}} & \multicolumn{1}{|c}{true} & false \\ \cline{2-4}
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\multirow{2}{*}{\textbf{A}} & \multicolumn{1}{l|}{true} & \multicolumn{1}{c}{\textcolor{mygreen}{true}} & \textcolor{red}{false} \erb \\
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& \multicolumn{1}{l|}{false} & \multicolumn{1}{l}{\textcolor{red}{false}} & \textcolor{red}{false}
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\end{tabular}
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\hfill
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\begin{tabular}{llll}
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\multicolumn{2}{l}{\multirow{2}{*}{}} & \multicolumn{2}{c}{\textbf{B}} \\
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& \sffamily{\textbf{oder}} & \multicolumn{1}{|c}{wahr} & falsch \\ \cline{2-4}
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\multirow{2}{*}{\textbf{A}} & \multicolumn{1}{l|}{wahr} & \multicolumn{1}{c}{\textcolor{mygreen}{wahr}} & \textcolor{mygreen}{wahr} \erb \\
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& \multicolumn{1}{l|}{falsch} & \multicolumn{1}{l}{\textcolor{mygreen}{wahr}} & \textcolor{red}{falsch}
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& \sffamily{\textbf{oder}} & \multicolumn{1}{|c}{true} & false \\ \cline{2-4}
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\multirow{2}{*}{\textbf{A}} & \multicolumn{1}{l|}{true} & \multicolumn{1}{c}{\textcolor{mygreen}{true}} & \textcolor{mygreen}{true} \erb \\
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& \multicolumn{1}{l|}{false} & \multicolumn{1}{l}{\textcolor{mygreen}{true}} & \textcolor{red}{false}
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\end{tabular}
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\hfill
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\begin{tabular}{llll}
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\multicolumn{2}{l}{\multirow{2}{*}{}} & \multicolumn{2}{c}{\textbf{B}} \\
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& \sffamily{\textbf{xor}} & \multicolumn{1}{|c}{wahr} & falsch \\ \cline{2-4}
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\multirow{2}{*}{\textbf{A}} & \multicolumn{1}{l|}{wahr} & \multicolumn{1}{c}{\textcolor{red}{falsch}} & \textcolor{mygreen}{wahr} \erb \\
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& \multicolumn{1}{l|}{falsch} & \multicolumn{1}{l}{\textcolor{mygreen}{wahr}} & \textcolor{red}{falsch}
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& \sffamily{\textbf{xor}} & \multicolumn{1}{|c}{true} & false \\ \cline{2-4}
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\multirow{2}{*}{\textbf{A}} & \multicolumn{1}{l|}{true} & \multicolumn{1}{c}{\textcolor{red}{false}} & \textcolor{mygreen}{true} \erb \\
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& \multicolumn{1}{l|}{false} & \multicolumn{1}{l}{\textcolor{mygreen}{true}} & \textcolor{red}{false}
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\end{tabular}
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\end{table}
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Tabelle \ref{logicalrelationaloperators} zeigt die logischen Operatoren, die in
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\matlab{} definiert sind. Zu bemerken sind hier noch die
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\code[Operator!logischer!and2@\&\&]{\&\&} und
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\code[Operator!logischer!or2@{"|}{"|} {}]{||} Operatoren. Man kann
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beliebige Ausdr\"ucke verkn\"upfen und h\"aufig kann schon anhand des
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ersten Ausdrucks entschieden werden, ob der gesamte boolesche Ausdruck
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zu wahr oder falsch ausgewertet werden wird. Wenn zwei Aussagen mit
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einem UND verkn\"upft werden und der erste zu falsch ausgewertet wird,
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muss der zweite gar nicht mehr gepr\"uft werden. Die Verwendung der
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\enterm{short-circuit} Versionen spart Rechenzeit, da die Ausdr\"ucke
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nur sowei wie n\"otig ausgewertet werden.
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Table~\ref{logicalrelationaloperators} show the logical and relational
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operators that are available in \matlab{}. The additional
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\code[Operator!logical!and2@\&\&]{\&\&} und
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\code[Operator!logical!or2@{"|}{"|} {}]{||} operators are the so
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called `\enterm{shorrt-circuit} operators for the logical OR and
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AND. Short-circuit means that \matlab{} stopps to evaluate a Boolean
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expresssion as soon as it becomes clear that the whole expression
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cannot become true. For example assume that the two statements A and B
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are joined using a AND. The whole expression can only be true if A is
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already true. This means, that there is no need to evaluate B if A is
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false. Since the statements may be arbitrarily elaborated computations
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this saves processing time.
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\begin{table}[t]
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\titlecaption{\label{logicalrelationaloperators}
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Logische (links) und relationale (rechts) Operatoren in \matlab.}{}
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Logical (left) and relational (right) operators in \matlab.}{}
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\begin{tabular}{cc}
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\hline
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\textbf{Operator} & \textbf{Beschreibung} \erh \\ \hline
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\varcode{$\sim$} & logisches NICHT \erb \\
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\varcode{$\&$} & logisches UND\\
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\varcode{$|$} & logisches ODER\\
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\varcode{$\&\&$} & short-circuit logisches UND\\
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\varcode{$\|$} & short-circuit logisches ODER\\
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\textbf{operator} & \textbf{description} \erh \\ \hline
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\varcode{$\sim$} & logical NOT \erb \\
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\varcode{$\&$} & logical AND\\
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\varcode{$|$} & logical OR\\
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\varcode{$\&\&$} & short-circuit logical AND\\
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\varcode{$\|$} & short-circuit logical OR\\
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\hline
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\end{tabular}
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\hfill
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\begin{tabular}{cc}
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\hline
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\textbf{Operator} & \textbf{Beschreibung} \erh \\ \hline
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\varcode{$==$} & gleich \erb \\
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\varcode{$\sim=$} & ungleich\\
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\varcode{$>$} & gr\"o{\ss}er als \\
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\varcode{$<$} & kleiner als \\
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\varcode{$>=$} & gr\"o{\ss}er oder gleich \\
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\varcode{$<=$} & kleiner oder gleich \\
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\textbf{operator} & \textbf{description} \erh \\ \hline
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\varcode{$==$} & equals \erb \\
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\varcode{$\sim=$} & unequal\\
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\varcode{$>$} & greater than \\
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\varcode{$<$} & less than \\
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\varcode{$>=$} & greateror equal \\
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\varcode{$<=$} & less or equal \\
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\hline
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\end{tabular}
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\end{table}
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Um Werte miteinander zu vergleichen gibt es die
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\codeterm[Operator!relationaler]{relationalen Operatoren} (Tabelle
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\ref{logicalrelationaloperators}). Mit ihnen kann man auf Dinge wie
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Gleichheit (\varcode{==}) gr\"o{\ss}er oder kleiner als (\varcode{>},
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\varcode{<}) testen.
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\begin{important}[Zuweisungs- und Gleichheitsoperator]
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Der Zuweisungsoperator \code[Operator!Zuweisung!=]{=} und der
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logische Operator \code[Operator!logischer!==]{==} sind zwei
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grundverschiedene Dinge. Da sie umgangsprachlich gleich sind
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k\"onnen sie leider leicht verwechselt werden.
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\begin{important}[Assignment and equality operators]
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The assignment operator \code[Operator!Assignment!=]{=} and the
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logical equality operator \code[Operator!logical!==]{==} are
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fundamentally different. Since they are colloquially treated equal
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they can be easily confused.
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\end{important}
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Das Ergebnis eines booleschen Ausdrucks ist immer vom Datentyp
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\codeterm{logical}. Jede beliebige Variable zu wahr oder falsch
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ausgewertet werden indem diese in den Typ \code{logical} umgewandelt
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wird. Dabei werden von \matlab{} alle Werte, die nicht 0 sind als wahr
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eingesch\"atzt. Listing \ref{booleanexpressions} zeigt einige
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Beispiele. \matlab{} kennt die Schl\"usselworte \code{true} und
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\code{false}. Diese sind Synonyme f\"ur die \code{logical} Werte 1 und
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0.
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\begin{lstlisting}[caption={Boolesche Ausdr\"ucke.}, label=booleanexpressions]
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Previously we have introduced the data types for integer or floating
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point numbers and discussed that there are instances where it is more
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efficient to use a integer data type rather than storing floating
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poing numbers. The result of a Boolean expression can only assume two
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values (true or false). This implies that we need only a single bit to
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store this information as a 0 (false) and 1 (true). In \matlab{} knows
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a special data type (\codeterm{logical}) to store the result of a
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Boolean expresssion. Every variable can be evaluated to true or false
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by just onverting it to the logical data type. When doing so \matlab{}
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interprets all values different form zero to be true. In
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listing~\ref{booleanexpressions} we show several examples for such
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operations. \matlab{} also knows the keywords \code{true} and
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\code{false} which are synonyms for the \codeterm{logical} values 1
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and 0.
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\begin{lstlisting}[caption={Boolean expressions.}, label=booleanexpressions]
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>> true
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ans = 1
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>> false
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