# bijections Published articles for bijections. 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. ## N Choose 2 is the Sum of the First N-1 Integers DevFeed: [N Choose 2 is the Sum of the First N-1 Integers]() Original publisher: [Read original article]() Published: 2011-10-02T16:16:47Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics]() Tags: [arithmetic](), [bijections](), [combinatorics](), [mathematics](), [proofs-without-words]() ### AI overview A combinatorial bijection shows that the binomial coefficient n choose 2 equals the sum of the first n−1 integers. The article explains how yellow dots correspond uniquely to pairs of dots in the bottom row, then briefly connects bijections to isomorphism and classification in mathematics. ### Source excerpt Problem: Determine an arithmetic expression for $ \binom{n}{2}$. Solution: The following picture describes a bijection between the set of yellow dots and the set of pairs of purple dots: In particular, selecting any yellow dots and travelling downward along diagonals gives a unique pair of blue dots. Conversely, picking any pair of blue dots gives a unique yellow dot which is the meeting point (the "peak") of the inward diagonals. If we say the bottom row has $ n$ elements, then the number of yellow dots is clearly $ 1 + 2 + \dots + (n-1)$, and the number of pairs in the last row is just $ \binom{n}{2}$. ## Set Theory--A Primer DevFeed: [Set Theory--A Primer]() Original publisher: [Read original article]() Published: 2011-07-09T18:14:59Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic]() Tags: [axiom-of-choice](), [bijections](), [cardinality](), [countability](), [example](), [functions](), [mathematics](), [numbers](), [power-set](), [primer](), [set](), [set-theory](), [symbols](), [theory](), [variable]() ### AI overview This primer introduces set theory by defining sets, elements, membership, cardinality, notation, and several ways to construct sets. It uses numerical examples and introduces natural numbers, integers, and rational numbers while noting that unrestricted operations can lead to paradoxes. ### Source excerpt It's often that a student's first exposure to rigorous mathematics is through set theory, as originally studied by Georg Cantor. This means we will not treat set theory axiomatically (as in ZF set theory), but rather we will take the definition of a set for granted, and allow any operation to be performed on a set. This will be clear when we present examples, and it will be clear why this is a bad idea when we present paradoxes.