- E0
- E0
\(β\) Show that addition is associative.
Solution 1We want to show that
\[ β n,m,kββ\quad ,\quad (n+m)+h=n+(m+h)\quad ; \]Define
\[ P(n)β β m,kββ\quad ,\quad (n+m)+h=n+(m+h)\quad ; \]Obviously \(P(0)\) is true; Study
\[ P(S(n))β β m,kββ\quad ,\quad (S(n)+m)+h\stackrel?=S(n)+(m+h)\quad ; \]we use the [(4.15)]
\begin{equation*} S(h) + n = S(h+n)= h + S(n) \end{equation*}Write
\begin{eqnarray*} (S(n)+m)+h \stackrel{\text{\scriptsize per \href{@URLPLACEHOLDER91HDLP@27P}{{\texttt{\scriptsize [({4.15})]}}}}}{=} S\Big((n+m)+h\Big) \stackrel{P(n)}{=} S\Big(n+(m+h)\Big) \stackrel{\text{\scriptsize per \href{@URLPLACEHOLDER91HDLP@27P}{{\texttt{\scriptsize [({4.15})]}}}}}{=} S(n)+(m+h) \end{eqnarray*}thus concluding the inductive step.
EDB β 27Y
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English
Authors:
"Mennucci , Andrea C. G."
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