# diagonalization

Published articles for diagonalization.

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## Methods of Proof -- Diagonalization

DevFeed: [Methods of Proof -- Diagonalization](<https://devfeed.tech/articles/methods-of-proof-diagonalization-40384.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2015/06/08/methods-of-proof-diagonalization/>)

Published: 2015-06-08T09:00:00Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Mathematics](<https://devfeed.tech/topics/mathematics.md>), [Math and Logic](<https://devfeed.tech/topics/math-and-logic.md>), [Collections](<https://devfeed.tech/topics/collections.md>)

Tags: [bijections](<https://devfeed.tech/tags/bijections.md>), [cardinality](<https://devfeed.tech/tags/cardinality.md>), [diagonalization](<https://devfeed.tech/tags/diagonalization.md>), [halting-problem](<https://devfeed.tech/tags/halting-problem.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [methods-of-proof](<https://devfeed.tech/tags/methods-of-proof.md>), [turing-machines](<https://devfeed.tech/tags/turing-machines.md>), [uncountability](<https://devfeed.tech/tags/uncountability.md>)

### AI overview

This tutorial introduces diagonalization as an advanced method of mathematical proof. It explains the table-and-diagonal construction and presents the theorem that no bijection exists between the natural numbers and the real numbers.

### Source excerpt

A while back we featured a post about why learning mathematics can be hard for programmers, and I claimed a major issue was not understanding the basic methods of proof (the lingua franca between intuition and rigorous mathematics). I boiled these down to the "basic four," direct implication, contrapositive, contradiction, and induction. But in mathematics there is an ever growing supply of proof methods. There are books written about the "probabilistic method," and I recently went to a lecture where the "linear algebra method" was displayed.

## Methods of Proof -- Contradiction

DevFeed: [Methods of Proof -- Contradiction](<https://devfeed.tech/articles/methods-of-proof-contradiction-40305.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/02/28/methods-of-proof-contradiction/>)

Published: 2013-02-28T11:46:02Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Mathematics](<https://devfeed.tech/topics/mathematics.md>), [Math and Logic](<https://devfeed.tech/topics/math-and-logic.md>), [Computing](<https://devfeed.tech/topics/computing.md>)

Tags: [bijections](<https://devfeed.tech/tags/bijections.md>), [countability](<https://devfeed.tech/tags/countability.md>), [diagonalization](<https://devfeed.tech/tags/diagonalization.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [methods-of-proof](<https://devfeed.tech/tags/methods-of-proof.md>), [proof-by-contradiction](<https://devfeed.tech/tags/proof-by-contradiction.md>), [theory](<https://devfeed.tech/tags/theory.md>)

### AI overview

This tutorial introduces proof by contradiction, explains how it is used to prove impossibility results, and applies the technique to a party-friends problem involving repeated numbers of friends. It also discusses functions on sets and different kinds of infinity.

### Source excerpt

In this post we'll expand our toolbox of proof techniques by adding the proof by contradiction. We'll also expand on our knowledge of functions on sets, and tackle our first nontrivial theorem: that there is more than one kind of infinity. Impossibility and an Example Proof by Contradiction Many of the most impressive results in all of mathematics are proofs of impossibility. We see these in lots of different fields. In number theory, plenty of numbers cannot be expressed as fractions.