# methods of proof Published articles for methods of proof. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Methods of Proof -- Diagonalization DevFeed: [Methods of Proof -- Diagonalization]() Original publisher: [Read original article]() Published: 2015-06-08T09:00:00Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic](), [Collections]() Tags: [bijections](), [cardinality](), [diagonalization](), [halting-problem](), [mathematics](), [methods-of-proof](), [turing-machines](), [uncountability]() ### 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]() Original publisher: [Read original article]() Published: 2013-02-28T11:46:02Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic](), [Computing]() Tags: [bijections](), [countability](), [diagonalization](), [mathematics](), [methods-of-proof](), [proof-by-contradiction](), [theory]() ### 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. ## Methods of Proof -- Contrapositive DevFeed: [Methods of Proof -- Contrapositive]() Original publisher: [Read original article]() Published: 2013-02-22T20:43:25Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [function](), [functions](), [object]() Tags: [contrapositive](), [function](), [functions](), [injective-functions](), [methods-of-proof](), [object](), [pigeonhole-principle](), [truth-tables]() ### AI overview This post introduces the contrapositive as one of four basic proof methods and begins developing the mathematical definition of functions on sets. It explains functions as sets of ordered pairs, with each input associated with exactly one output. ### Source excerpt In this post we'll cover the second of the "basic four" methods of proof: the contrapositive implication. We will build off our material from last time and start by defining functions on sets. Functions as Sets So far we have become comfortable with the definition of a set, but the most common way to use sets is to construct functions between them. As programmers we readily understand the nature of a function, but how can we define one mathematically? ## Methods of Proof -- Direct Implication DevFeed: [Methods of Proof -- Direct Implication]() Original publisher: [Read original article]() Published: 2013-02-16T12:56:53Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Math and Logic](), [Mathematics](), [Learning]() Tags: [contrapositive](), [direct-implication](), [induction](), [mathematics](), [methods-of-proof](), [set-theory](), [syntax](), [techniques]() ### AI overview This tutorial introduces a series on mathematical proof techniques for programmers and begins with direct implication. It presents four basic methods--direct implication, contradiction, contrapositive, and induction--and uses set theory to introduce notation and practice. ### Source excerpt I recently posted an exploratory piece on why programmers who are genuinely interested in improving their mathematical skills can quickly lose stamina or be deterred. My argument was essentially that they don't focus enough on mastering the basic methods of proof before attempting to read research papers that assume such knowledge. Also, there are a number of confusing (but in the end helpful) idiosyncrasies in mathematical culture that are often unexplained. ## Teaching Mathematics--Graph Theory DevFeed: [Teaching Mathematics--Graph Theory]() Original publisher: [Read original article]() Published: 2011-06-26T16:19:20Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [graph theory](), [Graphs](), [Mathematics]() Tags: [campolindo](), [education](), [google](), [graph-theory](), [introspection](), [mathematics](), [methods-of-proof](), [primer](), [puzzles](), [seven-bridges](), [seven-hills](), [teaching]() ### AI overview A mathematics teacher describes using puzzles and student-led discovery to introduce graph theory in high-school and middle-school lectures. The article presents the Seven Bridges of Königsberg problem as a central example and discusses mathematics as a non-linear process of questions, conjectures, and proof. ### Source excerpt Community Service Mathematics is supposed to be a process of discovery. Definitions, propositions, and methods of proof don't come from nowhere, although after the fact (when presented in a textbook) they often seem to. As opposed to a textbook, real maths is highly non-linear. It took mathematicians quite a lot of fuss to come up with the quadratic formula, and even simple geometric conjectures were for the longest time the subject of hot debate.