Questions tagged [induction]

For questions about mathematical induction, a method of mathematical proof. Mathematical induction generally proceeds by proving a statement for some integer, called the base case, and then proving that if it holds for one integer then it holds for the next integer. This tag is primarily meant for questions about induction over natural numbers but is also appropriate for other kinds of induction such as transfinite, structural, double, backwards, etc.

Mathematical induction is a form of deductive reasoning. Its most common use is induction over well-ordered sets, such as natural numbers or ordinals. While induction can be expanded to class relations which are well-founded, this tag is aimed mostly at questions about induction over natural numbers.

In general use, induction means inference from the particular to the general. This is used in terms such as inductive reasoning, which involves making an inference about the unknown based on some known sample. Mathematical induction is not true induction in this sense, but is rather a form of proof.

Induction over the natural numbers generally proceeds with a base case and an inductive step:

  • First prove the statement for the base case, which is usually $n=0$ or $n=1$.
  • Next, assume that the statement is true for an input $n$, and prove that it is true for the input $n+1$.

The following variant goes without a base case: Assuming the statement is true for all $n\in\mathbb N$ with $n < N$, prove that is true for $N$, too. This has to be done for all $N\in\mathbb N$.

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Proving the Existence of Triangle by Induction

There is an exercise which is should be proven by induction: $2n$ points are given in space. Altogether $n^2+1$ line segments are drawn between these points. Show that there is at least one set of three points which are joined pairwise by line…
com
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Non-induction proof of $2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1$

Prove that $$2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1.$$ After playing around with the sum, I couldn't get anywhere so I proved inequalities by induction. I'm however interested in solutions that don't use induction, if there…
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Discrete Math - Proving Distributive Laws for Sets by induction

I'm working on doing a proof by induction on this question: Use induction to prove that if $X_1, . . . , X_n$ and $X$ are sets, then $X∩(X_1∪X_2∪· · ·∪X_n) = (X∩X_1)∪(X∩X_2)∪· · ·∪(X∩X_n)$. I've shown the basis case: $X∩X_1 = X∩X_1$ But I'm having…
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Strong induction and vacuous truth

I was pondering a bit more about this question regarding being able to "omit" the base case in a proof by strong induction due to vacuous truth. The post states: Strong induction proves a sequence of statements $P(0), P(1), …$ by proving the…
rb612
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What is the $n$-time iterated adjugate of an $n\times n$ matrix $A$?

What is $\underbrace{\text{adj}\Big(\text{adj}\big(\ldots(\text{adj}}_{n\text{ adj}}\ A)\ldots\big)\Big)$, where $\text{adj}$ is written $n$ times, and the order of the matrix $A$ is $n\times n$? Can you show the proof for each $n$ (I mean by…
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Proof of $2^n > n$ by Induction

I'm new to induction and trying to prove $2^n > n$ for all natural numbers. I've seen a couple of examples but am confused about the the case going from $k = 1$ to $k =2$. So I show $2^1 > 1$ as the base case. Then I assume $2^k > k$ Meaning…
Robin Andrews
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Basic proof by Mathematical Induction (Towers of Hanoi)

I am new to proofs and I am trying to learn mathematical induction. I started working out a sample problem, but I am not sure if I am on the right track. I was wondering if someone would be kind enough to comment on my work so far, and give me some…
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Explain this derivative identity: $ \frac{1}{2^n} \frac{d^n}{dy^n} \frac{(1+y)^{2n+3}(1-y)}{((1+y)^2 -2yx)^2} \bigg|_{y=0} = (n+1)! x^n $

I have the following result that I believe to be true: $$ \frac{1}{2^n} \frac{d^n}{dy^n} \frac{(1+y)^{2n+3}(1-y)}{((1+y)^2 -2yx)^2} \bigg|_{y=0} = (n+1)! x^n $$ The LHS is something that arose in physics research. The RHS has been inferred by…
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Proof by Induction Question including Rational Numbers

I just recently covered 'rational numbers' in class and was assigned the following question to solve using induction for n, so that for all $q \in \mathbb{Q}$ \ {1}: $$\sum_{k=0}^n q^k = \frac{q^{n+1}-1}{q-1}$$ I am not entirely sure on where to…
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Proving the size of the unions of sets

How should one go about proving the following with induction? $$ \left| \bigcup_{i \in I} A_i \right| = \sum_{J \subseteq I} (-1)^{|J|+1} \left|\bigcap_{i \in J} A_i \right| $$ I is just a finite set, and $A_i$ is just any set within it.
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tree has exactly $k$ nodes with degree $4$. Show that this tree has $2k+2$ leaves.

Prove: If a tree has exactly $k \geq 1 $ nodes with degree $4$, then this tree has at least $2k +2 $ leaves. ( nodes with degree $< 4 $ are only allowed for the leaves ). So I think that we can solve this with induction. $ k = 1$ : So we have…
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Proving that first-order formulas can not distinguish sets of a certain size

Consider the set of first-order formulas over the empty signature, i.e. $\mathrm{FO}(\emptyset)$ (with variable set $Var$). The models over this signature are just characterized by their plain carrier set. Also, let $\mathrm{qr}(\phi)$ be the…
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Proving $5 \mid (n^5-n)$ for all $n \in \mathbb{Z}^+$

Prove for all $n \in \mathbb{Z}^+$ that $5 \mid (n^5-n)$ My proof Basis step: Since $5 \mid (1^5-1) \iff 5 \mid 0$ and $5 \mid 0$ is true, the statement is true for $n=1$. Inductive step: Assume the statement is true for $n = k$; that is, assume…
user502227
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Structural induction in haskell

Notation in this question derives from Haskell as follows. Let $[A]$ denote the collection of lists whose elements are drawn from the set $A$, so e.g. $[\mathbb{Z}]$ is the set of integer lists. Denote the empty list of any type by $[]$. Denote a…
Cadey Lewis
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Connectivity in a graph with even-degree vertices is preserved by edge removal

I was wondering if I could get some help with this proof. Essentially, I have to prove that when you have a connected graph with each vertex being of even degree (at least 2) and it is possible to go from any vertex to another through some path, if…
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