L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.
Exercice
Quelles sont les propriétés satisfaites par les relations internes suivantes ? Aucune justification n'est attendue.
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\(
\begin{array}{|c|}
\hline
S\\\hline
T\\\hline
A\\\hline
R\\\hline
\end{array}
\)
\(
\xymatrix{
x_{1} \ar@(lu,ru)[]\ar@/^0.79pc/[rdd] && x_{2} \\
&&\\
&x_{3} \ar@/^0.79pc/[luu]\ar@(ld,rd)[]&
}
\)
-
\(
\begin{array}{|c|}
\hline
S\\\hline
T\\\hline
A\\\hline
R\\\hline
\end{array}
\)
\( \begin{array}{c||ccc} & x_{1} & x_{2} & x_{3} \\\hline\hline x_{1} & 1 & 0 & 0\\ x_{2} & 0 & 0 & 0\\ x_{3} & 0 & 0 & 1\end{array}\)
-
\(
\begin{array}{|c|}
\hline
S\\\hline
T\\\hline
A\\\hline
R\\\hline
\end{array}
\)
\( \begin{array}{c||cccc} & x_{1} & x_{2} & x_{3} & x_{4} \\\hline\hline x_{1} & 1 & 1 & 1 & 1\\ x_{2} & 0 & 1 & 1 & 1\\ x_{3} & 0 & 0 & 1 & 1\\ x_{4} & 0 & 0 & 0 & 1\end{array}\)
-
\(
\begin{array}{|c|}
\hline
S\\\hline
T\\\hline
A\\\hline
R\\\hline
\end{array}
\)
\( \mathcal{R}=\left\{\begin{array}{ccc}
(x_{1}, x_{1}) & (x_{2}, x_{1}) & (x_{2}, x_{2})\\ (x_{2}, x_{3}) & (x_{2}, x_{4}) & (x_{2}, x_{5})\\ (x_{3}, x_{1}) & (x_{3}, x_{2}) & (x_{3}, x_{3})\\ (x_{3}, x_{4}) & (x_{3}, x_{5}) & (x_{4}, x_{1})\\ (x_{4}, x_{2}) & (x_{4}, x_{4}) & (x_{4}, x_{5})\\ (x_{5}, x_{1}) & (x_{5}, x_{2}) & (x_{5}, x_{3})\\ (x_{5}, x_{4}) & (x_{5}, x_{5}) & \end{array}
\right\}\\ \text{ sur } X=\{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\}\)
-
\(
\begin{array}{|c|}
\hline
S\\\hline
T\\\hline
A\\\hline
R\\\hline
\end{array}
\)
\(
\xymatrix{
x_{1} \ar@(lu,ru)[]\ar@/^0.79pc/[rdd] && x_{2} \ar@(lu,ru)[] \\
&&\\
&x_{3} \ar@(ld,rd)[]&
}
\)
-
\(
\begin{array}{|c|}
\hline
S\\\hline
T\\\hline
A\\\hline
R\\\hline
\end{array}
\)
\( \begin{array}{c||ccc} & x_{1} & x_{2} & x_{3} \\\hline\hline x_{1} & 1 & 0 & 1\\ x_{2} & 0 & 1 & 0\\ x_{3} & 1 & 0 & 1\end{array}\)
-
\(
\begin{array}{|c|}
\hline
S\\\hline
T\\\hline
A\\\hline
R\\\hline
\end{array}
\)
\(
\xymatrix{
x_{1} && x_{2} \ar@/^0.579pc/[ll]\ar@(lu,ru)[]\ar@/^0.579pc/[lldd] \\
&&\\
x_{4} \ar@/^0.579pc/[uu]\ar@/^0.579pc/[rruu] && x_{3} \ar@/^6.579pc/[lluu]\ar@/^0.579pc/[uu]\ar@(ld,rd)[]\ar@/^0.579pc/[ll] \\
}
\)
-
\(
\begin{array}{|c|}
\hline
S\\\hline
T\\\hline
A\\\hline
R\\\hline
\end{array}
\)
\( \mathcal{R}=\left\{\begin{array}{ccc}
(x_{1}, x_{1}) & (x_{2}, x_{1}) & (x_{2}, x_{2})\\ (x_{2}, x_{3}) & (x_{2}, x_{4}) & (x_{3}, x_{1})\\ (x_{3}, x_{3}) & (x_{4}, x_{1}) & (x_{4}, x_{3})\\ (x_{4}, x_{4}) & \end{array}
\right\}\\ \text{ sur } X=\{x_{1}, x_{2}, x_{3}, x_{4}\}\)
-
\(
\begin{array}{|c|}
\hline
S\\\hline
T\\\hline
A\\\hline
R\\\hline
\end{array}
\)
\( \mathcal{R}=\left\{\begin{array}{ccc}
(x_{1}, x_{1}) & (x_{2}, x_{2}) & (x_{3}, x_{3})\\ \end{array}
\right\}\\ \text{ sur } X=\{x_{1}, x_{2}, x_{3}\}\)
-
\(
\begin{array}{|c|}
\hline
S\\\hline
T\\\hline
A\\\hline
R\\\hline
\end{array}
\)
\( \begin{array}{c||cccc} & x_{1} & x_{2} & x_{3} & x_{4} \\\hline\hline x_{1} & 1 & 0 & 0 & 0\\ x_{2} & 1 & 1 & 0 & 0\\ x_{3} & 1 & 1 & 1 & 0\\ x_{4} & 1 & 1 & 1 & 1\end{array}\)
Cliquer ici pour afficher la solution
Exercice
- \(
\begin{array}{|c|c|c|c|}
\hline
\color{green}{S}&\color{green}{T}&\color{red}{\not A}&\color{red}{\not R}
\\\hline
\end{array}
\)
- \(
\begin{array}{|c|c|c|c|}
\hline
\color{green}{S}&\color{green}{T}&\color{green}{A}&\color{red}{\not R}
\\\hline
\end{array}
\)
- \(
\begin{array}{|c|c|c|c|}
\hline
\color{red}{\not S}&\color{green}{T}&\color{green}{A}&\color{green}{R}
\\\hline
\end{array}
\)
- \(
\begin{array}{|c|c|c|c|}
\hline
\color{red}{\not S}&\color{red}{\not T}&\color{red}{\not A}&\color{green}{R}
\\\hline
\end{array}
\)
- \(
\begin{array}{|c|c|c|c|}
\hline
\color{red}{\not S}&\color{green}{T}&\color{green}{A}&\color{green}{R}
\\\hline
\end{array}
\)
- \(
\begin{array}{|c|c|c|c|}
\hline
\color{green}{S}&\color{green}{T}&\color{red}{\not A}&\color{green}{R}
\\\hline
\end{array}
\)
- \(
\begin{array}{|c|c|c|c|}
\hline
\color{red}{\not S}&\color{red}{\not T}&\color{red}{\not A}&\color{red}{\not R}
\\\hline
\end{array}
\)
- \(
\begin{array}{|c|c|c|c|}
\hline
\color{red}{\not S}&\color{green}{T}&\color{green}{A}&\color{green}{R}
\\\hline
\end{array}
\)
- \(
\begin{array}{|c|c|c|c|}
\hline
\color{green}{S}&\color{green}{T}&\color{green}{A}&\color{green}{R}
\\\hline
\end{array}
\)
- \(
\begin{array}{|c|c|c|c|}
\hline
\color{red}{\not S}&\color{green}{T}&\color{green}{A}&\color{green}{R}
\\\hline
\end{array}
\)