2021년 2월 17일 (수) 08:14에 Pythagoras0님의 편집

2021-02-17T08:14:59Z

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-26T12:21:20Z

메타데이터: 새 문단

Pythagoras0: /* 노트 */ 새 문단

2020-12-21T09:52:28Z

노트: 새 문단

새 문서

== 노트 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q178377 Q178377]
===말뭉치===
# In the Algebra world, mathematical induction is the first one you usually learn because it's just a set list of steps you work through.<ref name="ref_735bf750">[https://www.coolmath.com/algebra/19-sequences-series/09-mathematical-induction-01 Cool math Algebra Help Lessons]</ref>
# I was going to start out by officially stating "The Principle of Mathematical Induction"...<ref name="ref_735bf750" />
# Induction proof is a mathematical method of proving a set of formula or theory or series of natural numbers.<ref name="ref_4dc2f622">[https://www.math-only-math.com/induction-proof.html Examples on Math Induction Proof]</ref>
# Induction proof is used from the theory of mathematical induction which is similar to the incident of fall of dominoes.<ref name="ref_4dc2f622" />
# Similarly in induction proof for infinite series of n numbers set where P (n) is the set property, we do not need to prove the property for all natural numbers.<ref name="ref_4dc2f622" />
# In the second step we need to assume first that the property P (k) is true which is called induction hypothesis.<ref name="ref_4dc2f622" />
# The following are typical of results that can be proved by induction: 1.<ref name="ref_0f7b2602">[https://www.oxfordreference.com/view/10.1093/oi/authority.20110803100139949 mathematical induction]</ref>
# (n + 1)! – 1 for all natural numbers using the principles of mathematical induction.<ref name="ref_5243acf0">[https://byjus.com/maths/principle-of-mathematical-induction-learn-examples/ Principle of Mathematical Induction]</ref>
# What is meant by mathematical induction?<ref name="ref_5243acf0" />
# Mathematical induction is defined as a method, which is used to establish results for the natural numbers.<ref name="ref_5243acf0" />
# Generally, this method is used to prove the statement or theorem is true for all natural numbers Write down the two steps involved in the principles of mathematical induction?<ref name="ref_5243acf0" />
# We will verify (3.3) by mathematical induction .<ref name="ref_8d268ae4">[https://www.thefreedictionary.com/mathematical+induction mathematical induction]</ref>
# I will tell you how mathematical induction works very soon (and hopefully you can feel like Neo for a second) but let me tell you this first.<ref name="ref_51dc5751">[https://towardsdatascience.com/no-need-to-know-the-end-recursion-algorithm-and-mathematical-induction-5a9e4c747c3c No Need to Know the End: Recursive Algorithm and Mathematical Induction]</ref>
# When I have learned about a recursive algorithm recently, it reminded me of the time I learned mathematical induction.<ref name="ref_51dc5751" />
# what similarities I found between a recursive algorithm and mathematical induction and how they help me to implement the algorithm.<ref name="ref_51dc5751" />
# Theory and Applications shows how to find and write proofs via mathematical induction.<ref name="ref_d0505cce">[https://www.routledge.com/Handbook-of-Mathematical-Induction-Theory-and-Applications/Gunderson/p/book/9781138199019 Handbook of Mathematical Induction Theory and Applications]</ref>
# He then introduces ordinals and cardinals, transfinite induction, the axiom of choice, Zorn’s lemma, empirical induction, and fallacies and induction.<ref name="ref_d0505cce" />
# Mathematical induction is designed to prove statements like this.<ref name="ref_6dba37ca">[https://www.utica.edu/faculty_staff/xixiao/public/mat305/section-5.html Mathematical Induction]</ref>
# First we see that the natural number \(n\) does not start with \(1\) as we require in mathematical induction.<ref name="ref_6dba37ca" />
# What we did in the last paragraph is called the generalized strong induction.<ref name="ref_6dba37ca" />
# The word "generalized" means that the induction can start with any number and not just \(1\).<ref name="ref_6dba37ca" />
# We will argue by induction.<ref name="ref_492442db">[https://people.math.sc.edu/sumner/numbertheory/induction/Induction.html Mathematical Induction]</ref>
# Induction arguments don't always start with the case n = 1.<ref name="ref_492442db" />
# In that case we can use the slightly more general version of induction below.<ref name="ref_492442db" />
# Once guessed, most such properties can be verified by induction.<ref name="ref_492442db" />
# Mathematical Induction is a mathematical proof method that is used to prove a given statement about any well-organized set.<ref name="ref_946ef1f8">[https://www.geeksforgeeks.org/principle-of-mathematical-induction/ Principle of Mathematical Induction]</ref>
# We can compare mathematical induction to falling dominoes.<ref name="ref_946ef1f8" />
# Well, yes, math is deductive and, in fact, mathematical induction is actually a deductive form of reasoning; if that doesn't make your brain hurt, it should.<ref name="ref_be510994">[https://www.siue.edu/~jloreau/courses/math-223/notes/sec-induction.html Mathematical Induction]</ref>
# I like to think of mathematical induction via an analogy.<ref name="ref_be510994" />
# The cool thing about induction (we will henceforth drop the formality of “mathematical induction”) is that it allows us to prove infinitely many statements.<ref name="ref_be510994" />
# Induction also allows us to define infinitely many things at the same time.<ref name="ref_be510994" />
# In this definitive guide to Mathematical Induction, I start from the beginning: precisely what is Mathematical Induction.<ref name="ref_d320a02d">[https://www.dave4math.com/mathematics/mathematical-induction/ Mathematical Induction (Theory and Examples)]</ref>
# I then work through examples using Strong Induction.<ref name="ref_d320a02d" />
# Towards, the end I cover arithmetic and geometric progressions as further examples of using induction.<ref name="ref_d320a02d" />
# Have you ever wondered why mathematical induction is a valid proof technique?<ref name="ref_d320a02d" />
# In this tutorial we’ll break down a classic induction problem in mathematics, and in the next post we’ll apply the same techniques to a classic computer science problem.<ref name="ref_88ded734">[http://blog.cambridgecoaching.com/what-is-mathematical-induction-and-how-do-i-use-it What is Mathematical Induction (and how do I use it?)]</ref>
# Mathematical Induction Mathematical induction is a common method for proving theorems about the positive integers, or just about any situation where one case depends on previous cases.<ref name="ref_e3170b24">[http://www2.edc.org/makingmath/mathtools/induction/induction.asp Making Mathematics: Mathematics Tools: Mathematical Induction]</ref>
# A slight variation on the induction hypothesis can be useful: assume that for all integers k < n your conjecture holds.<ref name="ref_e3170b24" />
# Mathematical induction is not only useful for proving algebraic identities.<ref name="ref_e3170b24" />
# See the second example below for a geometric application of induction.<ref name="ref_e3170b24" />
# In these cases it is convenient to use the following equivalent form of the principle of mathematical induction.<ref name="ref_eb85636b">[https://encyclopediaofmath.org/wiki/Mathematical_induction Encyclopedia of Mathematics]</ref>
# In these cases one has to deal with the proof of a number of assertions by compound mathematical induction.<ref name="ref_eb85636b" />
# Thus, a great number of ideas defined by compound mathematical induction lead to the need for an application of the axiomatic method in inductive definitions and proofs.<ref name="ref_eb85636b" />
# Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers.<ref name="ref_9ae24aee">[https://courses.lumenlearning.com/boundless-algebra/chapter/mathematical-inductions/ Mathematical Inductions]</ref>
# Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers (non-negative integers ).<ref name="ref_9ae24aee" />
# This completes the induction step.<ref name="ref_9ae24aee" />
# If you've done proof by induction before you may have been asked to assume the n-1 case and show the n case, or assume the n case and show the n+1 case.<ref name="ref_20ed3d9a">[http://comet.lehman.cuny.edu/sormani/teaching/induction.html Proof by Induction]</ref>
# Now I start with the left side of the equation I want to show and proceed using the induction hypothesis and algebra to reach the right side of the equation.<ref name="ref_20ed3d9a" />
# This is a different kind of proof by induction because it doesn't make sense until n=3.<ref name="ref_20ed3d9a" />
# Pk P(k+1) also satisfy the conditions of the induction hypothesis so we know Pj is between P1 and P(k+1) for any j=3...<ref name="ref_20ed3d9a" />
# Mathematical induction, is a technique for proving results or establishing statements for natural numbers.<ref name="ref_ec214677">[https://www.tutorialspoint.com/discrete_mathematics/discrete_mathematical_induction.htm Mathematical Induction]</ref>
# In problem solving, mathematical induction is not only a means of proving an existing formula, but also a powerful methodology for finding such formulas in the first place.<ref name="ref_fbc3a89b">[https://www.cut-the-knot.org/induction.shtml Mathematical Induction]</ref>
# Mathematical Induction is a special way of proving things.<ref name="ref_a736982d">[https://www.mathsisfun.com/algebra/mathematical-induction.html Mathematical Induction]</ref>
# Mathematical induction can be used to prove that an identity is valid for all integers \(n\geq1\).<ref name="ref_065bef24">[https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Book%3A_A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/03%3A_Proof_Techniques/3.04%3A_Mathematical_Induction_-_An_Introduction 3.4: Mathematical Induction - An Introduction]</ref>
# It turns out that we cannot completely prove the principle of mathematical induction with just the usual properties for addition and multiplication.<ref name="ref_065bef24" />
# Although we cannot provide a satisfactory proof of the principle of mathematical induction, we can use it to justify the validity of the mathematical induction.<ref name="ref_065bef24" />
# Therefore, the principle of mathematical induction proves that \(S=\mathbb{N}\).<ref name="ref_065bef24" />
# Let's go back to our example from above, about sums of squares, and use induction to prove the result.<ref name="ref_25fa797e">[https://nrich.maths.org/4718 An Introduction to Mathematical Induction]</ref>
# For example, if you're trying to sum a list of numbers and have a guess for the answer, then you may be able to use induction to prove it.<ref name="ref_25fa797e" />
# But you can't use induction to find the answer in the first place.<ref name="ref_25fa797e" />
# Subscribe today The foregoing is an example of simple induction; an illustration of the many more complex kinds of mathematical induction is the following method of proof by double induction.<ref name="ref_30ed5fde">[https://www.britannica.com/science/mathematical-induction mathematical induction | Definition, Principle, & Proof]</ref>
# Giuseppe Peano included the principle of mathematical induction as one of his five axioms for arithmetic.<ref name="ref_30ed5fde" />
# is to take it as a special case of transfinite induction.<ref name="ref_30ed5fde" />
# For example, there is a sense in which simple induction may be regarded as transfinite induction applied to the domain D of positive integers.<ref name="ref_30ed5fde" />
# And the way I'm going to prove it to you is by induction.<ref name="ref_f8bcd07a">[https://www.khanacademy.org/math/algebra-home/alg-series-and-induction/alg-induction/v/proof-by-induction Proof of finite arithmetic series formula by induction (video)]</ref>
# The way you do a proof by induction is first, you prove the base case.<ref name="ref_f8bcd07a" />
# So if we know it is true for 1 in our base case then the second step, this induction step must be true for 2 then.<ref name="ref_f8bcd07a" />
# Now spoken in generalaties let's actually prove this by induction.<ref name="ref_f8bcd07a" />
# A proof by induction consists of two cases.<ref name="ref_fec4839a">[https://en.wikipedia.org/wiki/Mathematical_induction Mathematical induction]</ref>
# Mathematical induction in this extended sense is closely related to recursion.<ref name="ref_fec4839a" />
# None of these ancient mathematicians, however, explicitly stated the induction hypothesis.<ref name="ref_fec4839a" />
# The first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique (1665).<ref name="ref_fec4839a" />
# Mathematical induction is a technique for proving a statement -- a theorem, or a formula -- that is asserted about every natural number.<ref name="ref_ee4b7414">[https://themathpage.com/aPreCalc/mathematical-induction.htm Mathematical induction]</ref>
# This is called the principle of mathematical induction.<ref name="ref_ee4b7414" />
# To prove a statement by induction, we must prove parts 1) and 2) above.<ref name="ref_ee4b7414" />
# k" -- is called the induction assumption, or the induction hypothesis.<ref name="ref_ee4b7414" />
===소스===
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 * ID :  [https://www.wikidata.org/wiki/Q178377 Q178377]

← 이전 판		2020년 12월 26일 (토) 12:21 판
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		+	* ID : [https://www.wikidata.org/wiki/Q178377 Q178377]

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2021년 2월 17일 (수) 08:14에 Pythagoras0님의 편집

Pythagoras0: /* 메타데이터 */ 새 문단

Pythagoras0: /* 노트 */ 새 문단