Explanation: Indicator Function and Complete Induction

(related to Part: Set-theoretic Prerequisites Needed For Combinatorics)

The indicator function provides new insight into the proving principle by complete induction. If $p$ is a property which is valid for some natural numbers $m\in\mathbb N,$ then it defines a unique indicator function $\chi(m)$ defined by $$\chi(m):=\begin{cases}1&\text{if }p(m)\text{ is true}\\0&\text{else, i.e if }p(m)\text{ is false.}\end{cases}$$ With this interpretation, the proving principle goes as follows: * Base case: Prove that $\chi(m)=1$ for some $m.$ * Induction step: Show that if $n\ge m$ and $\chi(m)=1$, then $\chi(n+1)=\chi(n).$ * Conclusion: Conclude that $\chi(n)=1$ for all $n\ge m.$

In other words, the proving principle uses the fact that the indicator function $\chi$ is constant locally, i.e. for all the two natural numbers $(n,n+1),$ if $n\ge m$ and $\chi(m)=\chi(n)=1.$ This allows the conclusion, that $\chi$ is constant globally, i.e. for all consecutive pairs $(n,n+1),$ $(n+1,n+2),$ etc, we have $\chi(m)=\chi(n)=\chi(n+1)=\chi(n+2)=\ldots.$ This, in turn, is equivalent to proving that the property $p$ is true for all natural numbers $n\ge m.$

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  1. Flachsmeyer, Jürgen: "Kombinatorik", VEB Deutscher Verlag der Wissenschaften, 1972, 3rd Edition