diff --git a/info4/kapitel-0/Logik.ipynb b/info4/kapitel-0/Logik.ipynb index d45bf0aaa1e7791f7beab1148fddfff233a634f4..a9692f78a17eaacdc7eb3278eaa2d6fd27e399d6 100644 --- a/info4/kapitel-0/Logik.ipynb +++ b/info4/kapitel-0/Logik.ipynb @@ -829,6 +829,60 @@ " NUND=bool((np=TRUE) ∧ (nq=TRUE))}" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Kontraposition\n", + "\n", + "$\\neg \\psi \\Rightarrow \\neg \\phi$ ist die Kontraposition von $\\phi \\Rightarrow \\psi$.\n", + " \n", + "Ist diese Form äquivalent?\n", + "Zur Prüfung kann man die komplette Wahrheitstabelle aufbauen, mit allen möglichen Werten für die \"offenen\" Formeln $\\phi$, $\\psi$" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "data": { + "text/markdown": [ + "|p|q|IMPL|KONT|np|nq|\n", + "|---|---|---|---|---|---|\n", + "|$\\mathit{FALSE}$|$\\mathit{FALSE}$|$\\mathit{TRUE}$|$\\mathit{TRUE}$|$\\mathit{TRUE}$|$\\mathit{TRUE}$|\n", + "|$\\mathit{FALSE}$|$\\mathit{TRUE}$|$\\mathit{TRUE}$|$\\mathit{TRUE}$|$\\mathit{TRUE}$|$\\mathit{FALSE}$|\n", + "|$\\mathit{TRUE}$|$\\mathit{FALSE}$|$\\mathit{FALSE}$|$\\mathit{FALSE}$|$\\mathit{FALSE}$|$\\mathit{TRUE}$|\n", + "|$\\mathit{TRUE}$|$\\mathit{TRUE}$|$\\mathit{TRUE}$|$\\mathit{TRUE}$|$\\mathit{FALSE}$|$\\mathit{FALSE}$|\n" + ], + "text/plain": [ + "p\tq\tIMPL\tKONT\tnp\tnq\n", + "FALSE\tFALSE\tTRUE\tTRUE\tTRUE\tTRUE\n", + "FALSE\tTRUE\tTRUE\tTRUE\tTRUE\tFALSE\n", + "TRUE\tFALSE\tFALSE\tFALSE\tFALSE\tTRUE\n", + "TRUE\tTRUE\tTRUE\tTRUE\tFALSE\tFALSE\n" + ] + }, + "execution_count": 2, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + ":table {p,q,IMPL,KONT,np,nq| p:BOOL & q:BOOL & np = bool(¬(p=TRUE)) & nq = bool(¬(q=TRUE)) &\n", + " IMPL=bool(p=TRUE => q=TRUE) &\n", + " KONT=bool((nq=TRUE) => (np=TRUE))}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Es gilt also:\n", + "$\\phi \\Rightarrow \\psi$ $\\equiv$ $\\neg \\psi \\Rightarrow \\neg \\phi$ ." + ] + }, { "cell_type": "markdown", "metadata": {},