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V%8TRZqEJVTA0Iuw{@BIw{{=A0?U?`o diff --git a/Terzo anno/Elementi di Teoria degli Insiemi/main.tex b/Terzo anno/Elementi di Teoria degli Insiemi/main.tex index f3b7d9f..7ec918e 100644 --- a/Terzo anno/Elementi di Teoria degli Insiemi/main.tex +++ b/Terzo anno/Elementi di Teoria degli Insiemi/main.tex @@ -1209,13 +1209,91 @@ originale in una formula esplicita: \item $x < y$, allora $\forall z (z \in N \implies x+z x$, e dunque $z > x$ per ipotesi induttiva, ovverosia + esiste $z''$ non nullo tale per cui $z = x + z''$. Poiché + $z''$ è non nullo, esiste $m$ tale per cui $z'' = m + 1$. Quindi + $z = x + (m + 1) = (x + 1) + m$. Se fosse $m = 0$, si avrebbe + $(x + 1) + y' = y + z' = z = x + 1$, \Lightning. Dunque + $z > x + 1$. + \end{enumerate} + + \item[\fbox{Confrontabilità}] Per induzione su $x$ con la formula + $P(x) = \forall y( x < y \lor x = y \lor x > y)$. + + \begin{enumerate} + \item[$\boxed{P(0)}$] Se $y \neq 0$, allora $0 < y$ dal momento che + $y = 0 + y$. + + \item[$\boxed{P(x + 1)}$] Per ipotesi induttiva, vale $y < x$, $y = x$ o + $y > x$. Se vale $y < x$, allora, per transitività, $y < x + 1$. Se + $y = x$, allora banalmente $y < x + 1$. Se invece $y > x$, esiste + $y'$ non nullo tale per cui $y = x + y'$. Poiché $y'$ non è nullo, allora + esiste $t$ tale per cui $y' = t + 1$. Dunque $y = x + (t + 1) = + (x + 1) + t$. Se $t = 0$, $y = x + 1$; altrimenti $y > x + 1$. + \end{enumerate} + \end{enumerate} + + \item Se $y > x$, allora esiste $y'$ non nullo per cui $y = x + y''$. Allora: + \[ y + z = (x + y'') + z = (x + z) + y'', \] + e quindi $y + z > x + z$. + + \item Se $y > x$, allora esiste $y'$ non nullo per cui $y = x + y''$. Allora: + \[ y \cdot z = (x + y'') \cdot z = x \cdot z + y'' \cdot z, \] + e quindi, poiché $y'' \cdot z$ (sia $y''$, che $z$ sono non nulli), + $y \cdot z > x \cdot z$. + + \item Mostriamo le due implicazioni. + + \begin{enumerate} + \item[$\boxed{n \in m \implies n < m}$] Se $n \in m$, allora $m = n \sqcup \underbrace{\{n, \ldots, m-1\}}_{=A}$. + Allora $m = n + \abs{A}$, e $\abs{A} \neq 0$ essendo $A$ non vuoto. Dunque $n < m$. + + \item[$\boxed{n \in m \impliedby n < m}$] Usiamo il fatto che $\in$ è un ordine totale su $\omega$. + Se $n \in m$, si conclude. Non si può avere $n = m$ per antiriflessività. Se fosse + $m \in n$, si avrebbe per l'implicazione inversa $m < n$, e quindi $m < m$, ancora contro + l'antiriflessività. + \end{enumerate} + \end{enumerate} +\end{solution} + \begin{problem}{$(\omega, \in)$ è un insieme ben ordinato}{problem-33} Si dimostri che $(\omega, \in)$ è un insieme ben ordinato. \end{problem}