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Proofs by Induction
No discussion of recursion and induction would be complete without some obligatory examples of proofs using induction. We start with the Hello World example.
Proofs by Induction
No discussion of recursion and induction would be complete without some obligatory examples of proofs using induction. We start with the Hello World example.
For our first version of a proof of , we clearly identify the open statement \(S_n\) and describe the proof carefully in terms of \(S_n\). As you develop more experience with writing proofs by induction, this will become less essential, as you'll see in the second version of the proof.
Proof
Let \(n\) be a positive integer, and let \(S_n\) be the open statement \[\begin{aligned}\end{aligned}\] We will prove that \(S_n\) is true for all positive integers by induction. For the basis step, we must prove that \(S_1\) is true. When \(n=1\), the left-hand side of \(S_n\) is just \(1\), while the right-hand side evaluates to \(1(1+1)/2=1\). Therefore, \(S_1\) is true.
Next we assume that for some positive integer \(k\), \(S_k\) is true. That is, we assume \[\begin{aligned}\end{aligned}\] We now seek to prove that \(S_{k+1}\) is true, and begin by considering the left-hand side of \(S_{k+1}\). We notice that \[\begin{aligned}\end{aligned}\] since our inductive hypothesis that \(S_k\) is true gives us the simpler formula for the summation. Now continuing with a bit of algebra, we find \[\begin{aligned}\end{aligned}\] Therefore, \(S_{k+1}\) is true. Since we have shown that \(S_1\) is true and that for every positive integer \(k\), if \(S_k\) is true, then \(S_{k+1}\) is true, we conclude that \(S_n\) is true for all positive integers \(n\) by the .
Before looking at a refined version of this proof, let's take a moment to discuss the key steps in every proof by induction. The first step is the basis step, in which the open statement \(S_1\) is shown to be true. (It's worth noting that there's nothing special about \(1\) here. If we want to prove only that \(S_n\) is true for all integers \(n\geq 5\), then proving that \(S_5\) is true is our basis step.) When proving the basis step, if \(S_n\) is an equation, we do not just write down \(S_1\) and move on. We need to prove that \(S_1\) is true. Notice how in the proof above, we discussed the left-hand side of \(S_1\) and the right-hand side of \(S_1\) and concluded that they were equal.
Now let's take a look at a more refined proof of . From here on, when we give a proof by induction, we'll use this style. As you're getting started with induction proofs, you may find it useful to be more explicit about the steps as we did in the first proof above.
Condensed: the full section is in Keller & Trotter, Applied Combinatorics.
À vous Aucune calculatrice ne tranche celle-ci, mais ses éléments sont calculables. Essayez-en un ci-dessous, ou tapez le vôtre.
Exercices (2)
Essayez chacun d'abord sur papier. Révélez la réponse pour vérifier; les vérifiés peuvent être ouverts dans le solveur pour chaque étape.
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Suppose that \(t(n)\) is a recursive function defined for \(n\geq 1\) such that \(t(1)=1\) and for \(n\geq 2\), \(t(n)=2t(n-1)+1\). Create a proof that for all \(n\geq 1\), \(t(n) = 2^n - 1\).
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Write a couple of sentences explaining why the making the inductive hypothesis is not the same as assuming what we are trying to prove.
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Inscription ConnexionSymboles utilisés ici
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Questions posées par les gens
Permutation or combination?
Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).
What is a graph in this sense?
Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.
Certaines parties de cette page sont adaptées de Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensés et réexpliqués ici ; les erreurs sont les nôtres.
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