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2項関係(17)

定義 injective relation

\begin{array}{l}R:{\mathop{\rm injective}\nolimits} \\\mathop  \equiv \limits^{def} {R^T}:{\mathop{\rm univalent}\nolimits} \\ \equiv R \circ {R^T} \subseteq {\rm I}\\ \equiv \overline {\rm I}  \circ R \subseteq \overline R \end{array}

定義 surjective relation

\begin{array}{l}R:{\mathop{\rm surjective}\nolimits} \\\mathop  \equiv \limits^{def} {R^T}:{\mathop{\rm total}\nolimits} \\ \equiv \Omega  = \Omega  \circ R\\ \equiv {\rm I} \subseteq {R^T} \circ R\\ \equiv \overline R  \subseteq \overline {\rm I}  \circ R\end{array}

定理

R:{\mathop{\rm surjective}\nolimits} iff 任意の関係Sに対し、R \circ S = \emptysetならばS = \emptyset

定義 bijective relation

\begin{array}{l}R:{\mathop{\rm bijective}\nolimits} \\\mathop  \equiv \limits^{def} R:{\mathop{\rm injective}\nolimits}  \wedge R:{\mathop{\rm surjective}\nolimits} \\ \equiv \overline {\rm I}  \circ R = \overline R \end{array}