EDB β€” 27Y

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E0

[27Y]

E0

\(⋆\) Show that addition is associative.

Solution 1

We want to show that

\[ βˆ€ n,m,kβˆˆβ„•\quad ,\quad (n+m)+h=n+(m+h)\quad ; \]

Define

\[ P(n)≐ βˆ€ m,kβˆˆβ„•\quad ,\quad (n+m)+h=n+(m+h)\quad ; \]

Obviously \(P(0)\) is true; Study

\[ P(S(n))≐ βˆ€ m,kβˆˆβ„•\quad ,\quad (S(n)+m)+h\stackrel?=S(n)+(m+h)\quad ; \]

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.

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