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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Must we use induction to prove a statement for all integers

This question is prompted by a remark from Bill Dubuque in his answer to this question on proving a particular sum without using mathematical induction. From Bill's answer: A proof that a statement is true for all integers must - at some point or…
user23784
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Prove that the 25 people can be seated in this way

5 mathematicians, 5 biologists, 5 chemists, 5 physicists, and 5 economists sit around a large round table. Prove that the 25 people can be seated such that, if A and B are two different people with the same specialty (for example, two…
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Can't find the flaw in the reasoning for this proof by induction?

I was looking over this problem and I'm not sure what's wrong with this proof by induction. Here is the question: Find the flaw in this induction proof. Claim $3n=0$ for all $n\ge 0$. Base for $n=0$, $3n=3(0)=0$ Assume Induction Hypothesis: $3k…
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Uses of "Collatz induction"?

The Collatz conjecture is equivalent to the following "induction principle": If $P(0) \land P(1) \land (\forall{x} P(3 \cdot x + 2) \implies P(2 \cdot x + 1)) \land (\forall x P(x) \implies P(2 \cdot x))$, then $\forall x P(x)$. I am wondering if…
Dan Brumleve
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Why is "mathematical induction" called "mathematical"?

One of my whims is that I never write "mathematical induction" but just "induction". We are doing maths, so what is the point about precising? We don't say "Let $f$ be a mathematical function from the mathematical set of mathematical real numbers…
Taladris
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Prove that the power set of an $n$-element set contains $2^n$ elements

Theorem. Let $X$ denote an arbitrary set such that $|X|=n$. Then $|\mathcal P(X)|=2^n$. Proof. The proof is by induction on the numbers of elements of $X$. For the base case, suppose $|X|=0$. Clearly, $X=\emptyset$. But the empty set is the only…
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What are some examples of induction where the base case is difficult but the inductive step is trivial?

According to Wikipedia: ...proofs by mathematical induction have two parts: the "base case" that shows that the theorem is true for a particular initial value such as n = 0 or n = 1 and then an inductive step that shows that if the…
Lucky
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How to prove a formula for the sum of powers of $2$ by induction?

How do I prove this by induction? Prove that for every natural number n, $ 2^0 + 2^1 + ... + 2^n = 2^{n+1}-1$ Here is my attempt. Base Case: let $ n = 0$ Then, $2^{0+1} - 1 = 1$ Which is true. Inductive Step to prove is: $ 2^{n+1} = 2^{n+2} - 1$…
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IMO 1987 - function such that $f(f(n))=n+1987$

Show that there is no function $f: \mathbb{N} \to \mathbb{N}$ such that $$f(f(n))=n+1987, \ \forall n \in \mathbb{N}$$ This is problem 4 from IMO 1987 - see, for example, AoPS.
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Why does induction have to be an axiom?

I noticed that there is an axiom that says that if $S(n)\implies S(n+1)$, and $S(1)$ is true, then $\forall n \in \Bbb N, S(n).$ My question is why is this an axiom? why can't we derive this from the other axioms?
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Why is mathematical induction a valid proof technique?

Context: I'm studying for my discrete mathematics exam and I keep running into this question that I've failed to solve. The question is as follows. Problem: The main form for normal induction over natural numbers $n$ takes the following…
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Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$

I am currently helping a friend of mine with his preperations for his next exam. A big topic of the exam will be induction, thus I told him he should practice this a lot. As at the beginning he had no idea how induction worked, I showed him some…
Huy
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How does one actually show from associativity that one can drop parentheses?

I've always heard this reasoning, and it makes obvious sense, but how do you actually show it for some arbitrary product? Would it be something like this? $$(a(b(cd)))e=((ab)(cd))e=(((ab)c)d)e=abcde?$$ Do you just say that the grouping of the…
user6927
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Vandermonde determinant by induction

For $n$ variables $x_1,\ldots,x_n$, the determinant $$ \det\left((x_i^{j-1})_{i,j=1}^n\right) = \left|\begin{matrix} 1&x_1&x_1^2&\cdots & x_1^{n-1}\\ 1&x_2&x_2^2&\cdots & x_2^{n-1} \\ \vdots&\vdots&\vdots&\ddots&\vdots\\ …
Alec Teal
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How do you prove Well-Ordering without Mathematical Induction? (and vice-versa)

Here is my attempt to prove the Well-Ordering Principle, i.e. that any non-empty subset of $\Bbb N$, the set of natural numbers, has a minimum element. Proof. Suppose there exists a non-empty subset $S$ of $\Bbb N$ such that $S$ has NO minimum…
Alexy Vincenzo
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