maths.free › Combinatorics & Graph Theory › Induction › Proofs by Induction

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.

Dabar tu Joks skaičiuotuvas nenusistato, bet jo dalys yra skaičiuojamos. Pabandykite vieną žemiau arba įveskite savo.

Praktika (2)

Išbandykite kiekvieną ant popieriaus pirmą. Atskleisti atsakymą patikrinti; patikrintas tie gali būti atidarytas sprendėjas už kiekvieną žingsnį.

  1. 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 \(n\geq 1\), \(t(n) = 2^n - 1\).

  2. Write a couple of sentences explaining why the making the inductive hypothesis is not the same as assuming what we are trying to prove.

Išlaikyti savo darbą

Nemokama paskyra prideda pastabas apie kiekvieną pamoką, įrašą apie tai, ką baigėte, savo išspręstas problemas vienoje vietoje ir mokytoją, kurį galite paklausti apie šį puslapį. Pati matematika yra atvira visiems, kurie yra ar nėra užsiregistravę.

Prisijungti Prisijungti

Čia naudojami simboliai

Bakstelėkite visus simbolius, kad būtų galima apibrėžti visą apibrėžimą, paveikslėlį ir ką reiškia kiekviena raidė.

Klausimai, kuriuos klausia žmonės

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.

Šio puslapio dalys pritaikytos nuo Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Čia yra įtikinamų ir iš naujo paaiškintų klaidų.

Daugiau informacijos Combinatorics & Graph Theory

The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs