Search results “Analysis an introduction to proof”

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Part 2 : https://youtu.be/V5tUc-J124s
In this video, I discuss 4 fundamental proof techniques used in mathematical proofs. I also talk about mathematical statements, logical operators and quantifiers.
This is part 1 of a video series discussing proofs.

Views: 26188
patrickJMT

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Part 1: https://youtu.be/KRLBya7x5ZQ
Extra Proof by Contradiction with some death intrigue (huh?!) https://www.youtube.com/watch?v=rOGqq1O1rzI&feature=youtu.be
New to proving mathematical statements and theorem? I this video I prove the statement 'the sum of two consecutive numbers is odd' using direct proof, proof by contradiction, proof by induction and proof by contrapositive.

Views: 136824
patrickJMT

First lecture in an introduction to real analysis. Topics include an overview of analysis, a review of logic, and an introduction to proof. This lecture is in four parts.

Views: 32707
ohubrismine

Real Analysis: We introduce some notions important to real analysis, in particular, the relationship between the rational and real numbers. Prerequisites may be found in the Math Major Basics playlist.

Views: 156991
MathDoctorBob

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Analysis is different from earlier mathematics such as calculus. And we need to learn new knowledge and skills. Let’s take a look at a typical page in an analysis book.
The left side box shows a typical page of analysis books. At first glance, this is frightening and boring. But it’s actually very exciting since we are going to learn some sophisticated mathematics.
This page contains many symbols and abbreviations. The right side box shows what each means and how to pronounce it. These symbols work as abbreviations such that we can express complex mathematical ideas in a condensed form. We should master them before doing anything else. They are very, very powerful.
‘a n’ is a general sequence. The right arrow means ‘tends to’ or ‘converges to’. This upside-down A means ‘for all’ or ‘for every’. This is the Greek letter epsilon, it is used as a variable. This reversed E means ‘there exists’ or ‘this is, there are’. This set membership symbol is read as ‘in’ or ‘is an element of’. This big N represents the natural numbers set, which includes 1, 2, 3, and so on. ‘max’ means ‘the maximum of’. And lastly, the curly brackets represents a set, here it is the set that contains the numbers N1 and N2.
Let’s look at the lecture note and identify its key components. It contains a definition, a theorem and a proof. The definition tells us what it means for a sequence to converge to a limit. The theorem tells what happens when we combine two convergent sequences ‘a n’ and ‘b n’ by multiplying their respective terms. The proof is an logical argument that concludes that the theorem is true. The proof first assumes that the two sequences ‘a n’ and ‘b n’ satisfy the definition, and then concludes that the multiplied sequence ‘a n times b n’ satisfies the definition too, thus proves the theorem.
The note might look like a jumble of symbols and words, but it can read aloud like any other sentences. Let’s try it.
Definition: A sequence an converges to a if and only if that for every number epsilon bigger than zero, there exists a natural number N such that for every number n greater than big N, the absolute value of ‘a n’ minus a is less than epsilon.
Theorem: Suppose that a sequence ‘a n’ converges to ‘a’, and another sequence ‘b n’ converges to ‘b’, then the sequence of the ‘a n times b n’ converges to ‘a times b’. The sequence ‘a n times b n’ is made by multiplying the corresponding ‘a n’ and ‘b n’ terms.
Proof: Let a sequence ‘a n’ converge to ‘a’, and the sequence ‘b n’ converge to b.
Let a variable epsilon greater than zero be arbitrary.
Then there exists a natural number ‘N 1’ such that for every ‘n’ greater than ‘N 1’, the absolute value of ‘a n minus a’ is less than epsilon over ‘2 times the absolute value of b plus 1’.
Also sequence ‘a n’ is bounded since all convergent sequences are bounded.
So there exists a number ‘M’ greater than zero, such that for all ‘n’ in the Natural Number set N, the absolute value of ‘a n’ is less or equal to ‘M’
For this ‘M’, there exits a number ‘N 2’ in the natural number set N, such that for all ‘n’ greater than ‘N 2’, the absolute value of ‘b n minus b’ is less than epsilon over ‘2 M’.
Let N be the maximum of the set ‘N 1 and N 2’.
Then for all ‘n’ greater than ‘N’, the absolute value of ‘a n times b n minus a times b’ is equal to the absolute value of ‘a n times b n minus a n times b plus a n times b minus a times b’.
Which is less than or equal to the absolute value of ‘a n times b n minus b’ plus the absolute value of b times a n minus a.
Which equals to the absolute value of ‘a n’ times the absolute value of ‘b n minus b’ plus the absolute value of ‘b’ times the absolute value of ‘a n minus a’.
Which is less than ‘M times epsilon over 2 M’ plus ‘the absolute value of b times epsilon over 2 times the absolute value of b plus 1’.
Which is less than epsilon over 2 plus epsilon over 2
Which equals epsilon.
Hence the sequence of ‘a n times b n’ converges to a times b.
Analysis is pure mathematics and should be understood as a theory, a network of general results liked together by valid logical arguments known as proofs. Analysis does not focus on repetitious calculations like calculus, but rather it focuses on the theory. It is the theorems, proofs and ways of thinking about them that you need to understand and master.

Views: 7661
FOSCO CONNECT

It's time to apply our new predicate logic to proofs and make use of all those new quantifiers. Also introduces mathematical induction.
http://www.polymathlectures.org/

Views: 36199
Bill Shillito

Real Analysis Proofs, Video #7 (Fundamental Inequalities V). Use formula for absolute value functions and basic lemmas.

Views: 7038
Bill Kinney

A very quick introduction to mathematical proofs.
Get live help with proofs at http://goo.gl/QH07IV.

Views: 9446
mathapptician

Proving an expression for the sum of all positive integers up to and including n by induction
Watch the next lesson: https://www.khanacademy.org/math/precalculus/seq_induction/proof_by_induction/v/alternate-proof-to-induction-for-integer-sum?utm_source=YT&utm_medium=Desc&utm_campaign=Precalculus
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Khan Academy

In this video, I share four research-based tips that can help you to teach proofs in mathematics classes.
This is my final project for EDUC695 class, "Research and Educational Practice", at the University of Michigan in Winter term 2017, taught by Nell Duke.
Studies cited in the video (in order they appear in the video) :
Sandefur, J., Mason, J., Stylianides, G. J., & Watson, A. (2013). Generating and using examples in the proving process. Educational Studies in Mathematics, 83(3), 323-340.
Weber, K., Alcock, L., & Radu, I. (2005). Undergraduates’ use of examples in a transition-to-proof course. In Proceedings of the 26th Conference for the North American Chapter of the Psychology of Mathematics Education. Electronic CD. Roanoke, VA.
Gibson, D. (1998). Students’ use of diagrams to develop proofs in an introductory real analysis. Research in Collegiate Mathematics Education, 2, 284–307.
Yousef, M., So, S., & Ewing, D. (2012). The modified Moore method versus the traditional: a case study. Far East Journal of Mathematical Education, 8(1).
Cooper, T. E., Bailey, B., & Briggs, K. (2012). The impact of a modified moore method on efficacy and performance in precalculus. Primus : Problems, Resources, and Issues in Mathematics Undergraduate Studies, 22(5), 386-410.
Marshall, J. C., & Horton, R. M. (2011). The relationship of teacher‐facilitated, inquiry‐based instruction to student higher‐order thinking. School Science and Mathematics, 111(3), 93-101.
Lee, K.: (2011). Students' Logical Reasoning and Mathematical Proving of Implications, Doctoral Dissertation, Michigan State University.
Contact me: [email protected]

Views: 13423
Ksenia Niglas

Topology Unit, Part 3. (0:00) Goals for the lecture. (0:28) Euclidean n-dimensional space. (5:00) Properties of metrics are satisfied by the Euclidean metric. (6:58) Sketch the proof of the triangle inequality for the Euclidean norm and then the Euclidean metric. (17:03) Defining a topology (define open and closed sets) in a metric space. (27:39) A reminder to make sure you think beyond the standard pictorial examples. What is the interior of the rationals? What is the derived set of the rationals? (29:07) Continue with the definition of a closed set. (30:59) The empty set is both open and closed (make the truth table for an implication). (33:52) The closure of a set. (34:49) Any open ball is an open set (and give a sketch of the proof, which requires the triangle inequality). (44:52) Open covers, finite subcovers, compact sets. (52:13) In Euclidean space, a set being compact is equivalent to being closed and bounded (Heine-Borel Theorem). (55:38) Why are compact sets important? Discuss this in terms of the Extreme Value Theorem.

Views: 5549
Bill Kinney

In this video we define the limit of a sequence, motivated by our intuitive understanding of the concept.
(PART 1/5 OF A 5 PART VIDEO SERIES)

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Q.E.D

How to find supremum infimum Real Analysis Mathematics

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maths gotserved

Introduction to convex sets in R^n.
Video created with Doce Nos http://bitly.com/Lx8UdN and iMovie.

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mathapptician

Taylor polynomials are incredibly powerful for approximations, and Taylor series can give new ways to express functions.
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3Blue1Brown

Math money, baby! In this important lesson, we introduce the concept of proofs in Geometry. We start with a quick run through of some common properties, then we solve 2 proofs. The first is an algebraic proof, then the second one is a brief one on segments. Im dressed as Money Math, so join and help me throw money at the problems. YAY MATH!
Please visit yaymath.org
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yaymath

Thanks for subscribing!
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This video is about proving time complexities ( big O ).
In the video the following concepts are explained:
- How to prove time complexities
- An example proof ( linear function )
- An example counterexample
- The definition of big-O
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MisterCode

How to give reasons to justify statements in proofs

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larryschmidt

In this tutorial I show how to do a proof by mathematical induction.
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We introduce proofs by looking at the most basic type of proof, a direct proof.
Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.

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TheTrevTutor

Sorry about the background noise and the wobbly camera view. (0:00) Lecture summary. (1:07) Statement of the Extreme Value Theorem (EVT). (6:17) Sketch of the proof that f is bounded on I = [a,b] (when f is continuous on I). (14:29) Sketch of the proof that there exists a c in I = [a,b] such that f(c) is the infimum (global minimum value) of the outputs. (21:31) Definition of the intermediate value property. (23:57) There are discontinuous functions that satisfy the intermediate value property. The function f with f(x) = x^2 * sin(1/x) when x is not 0 and f(0) = 0 is differentiable everywhere (including x = 0) but its derivative f' is not continuous at x = 0 (it does satisfy the intermediate value property on an interval containing zero, however...this is true in general for derivatives that exist everywhere, even when they are not continuous on the entire interval). (28:12) Mention the necessity that the interval is closed and bounded in the statement of the Extreme Value Theorem. (29:20) Definition of uniform continuity of a function over an interval. (31:25) Example: f(x) = x^2 is uniformly continuous on the open interval I = (2,7) (show scratch work for proof). (34:28) The Uniform Continuity Theorem (a function that is continuous on a closed and bounded interval [a,b] is also uniformly continuous on that same interval).

Views: 2205
Bill Kinney

Topology Unit, Part 2. (0:00) Final Exam schedule for Fall 2016. (0:49) Summary of sections to read and learn from Section 8.5. (1:36) The subject is topology. It's more concrete still in Lecture 32, but we will be getting more abstract in a discussion on metric spaces. (2:20) Plan for this lecture: we will focus on the real line R and the plane R^2. (5:04) Definition of an interior point of a set in both cases. (15:23) Definition of an open set in both cases. (22:01) Definition of a limit point of a set and a closed set in both cases. (42:52) Finite sets are closed because they don't have any limit points. The empty set is open and closed. The entire "ambient" space is both open and closed (in itself). (46:32) Definitions of: interior of a set, derived set (the set of limit points) of a set, and the closure of a set. The interior of a set is always open. The derived set and closure of a set are always closed. (53:36) Continuity and preimages of open sets of functions defined on the plane.

Views: 6891
Bill Kinney

The concepts of open and closed sets within a metric space are introduced

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Ben1994

In this video, we finish our proof of the existence of Euler's number.
We successfully prove that the sequence whose limit defines Euler's number is monotonic increasing.
(PART 5/5 OF A 5 PART SERIES OF VIDEOS)

Views: 237
Q.E.D

In this video we establish the least upper bound property of the real numbers. We then use this property in order to prove the Monotone Convergence Theorem.
(PART 2/5 OF A 5 PART VIDEO SERIES)

Views: 882
Q.E.D

More free math videos on mathgotserved.com thanks :D

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maths gotserved

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We take a look at a proof technique called Proof by Case
Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.

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TheTrevTutor

First lecture in an introduction to real analysis. Topics include an overview of analysis, a review of logic, and an introduction to proof. This lecture is in four parts.

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ohubrismine

We look at theorem 2.1.1 in Haggarty, and the proof thereof, with much emphasis on the desired analytical skills for students to appreciate the importance and tone of mathematical analysis.

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MATHSMANN

Join Crista Cowan for the fourth in our series on the Genealogical Proof Standard (GPS). Step three is "analysis and correlation of the collected information." You will learn both WHY this is such an important step in your research process and HOW to do it.
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Analysis and Correlation of Collected Information | Genealogical Proof Standard | Ancestry
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Ancestry

Moreras theorem proof in hindi.
Moreras theorem proof - complex analysis.
#morerastheorem #morerastheoreminhindi #morerastheoremproof #complexanalysis #MathematicsAnalysis
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Mathematics Analysis

We finish the characterization of the pre-compact subsets of the space of continuous functions. The pre-compact sets are found to be a family of functions with the attributes of uniform boundedness and equicontinuity.

Views: 1222
Arkady Etkin

Hey everyone, I am YourPalLucky. Sorry this video is kinda long compared to my other theory videos, I had a lot to talk about! Also sorry this is really late, I was at school when the teaser came out, and so much kept on coming out when I was working on this video. I hope you enjoy it though!
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YourPalLucky

This video is useful for students of BSc/MSc Mathematics students. Also for students preparing IIT-JAM, GATE, CSIR-NET and other exams.

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Bhagwan Singh Vishwakarma

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Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space. Accepting this not only gives us more rich and complete mathematics, but also unlocks a ridiculous amount of very real, very tangible problems in science and engineering.
Part 1: Introduction
Part 2: A Little History
Part 3: Cardan's Problem
Part 4: Bombelli's Solution
Part 5: Numbers are Two Dimensional
Part 6: The Complex Plane
Part 7: Complex Multiplication
Part 8: Math Wizardry
Part 9: Closure
Part 10: Complex Functions
Part 11: Wandering in Four Dimensions
Part 12: Riemann's Solution
Part 13: Riemann Surfaces
Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: http://www.welchlabs.com/resources.

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Welch Labs

Stanford University
http://openclassroom.stanford.edu/MainFolder/CoursePage.php?course=IntroToAlgorithms
Recommend Videos : https://youtu.be/yRM3sc57q0c
(1/4/2011)
CS 161 - Design and Analysis of Algorithms
Prof. Tim Roughgarden
COURSE DESCRIPTION :
Course Overview: Introduction to fundamental techniques for designing and analyzing algorithms, including asymptotic analysis; divide-and-conquer algorithms and recurrences; greedy algorithms; data structures; dynamic programming; graph algorithms; and randomized algorithms.
Required textbook: Kleinberg and Tardos, Algorithm Design, 2005. We will be covering most of Chapters 4–6, some parts of Chapter 13, and a couple of topics not in the book.
Prerequisites: Introduction to proofs, and discrete mathematics and probability (e.g., CS 103 and Stat116). If you have not taken a probability course, you should expect to do some independent reading during the course on topics including random variables, expectation, conditioning, and basic combinatorics.

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資訊107甲13邱大岡

In this video, we find bounds for the sequence whose limit defines Euler's number.
(PART 4/5 OF A 5 PART SERIES OF VIDEOS)

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Q.E.D

In this video we define convergence of an infinite series. We also realize that the definition of the convergence of an infinite series is not very different to the definition of the convergence of a infinite sequence which was discussed in the first video.
(PART 3/5 OF A 5 PART SERIES OF VIDEOS)

Views: 430
Q.E.D

Basic Methods: We note the different methods of informal proof, which include direct proof, proof by contradiction, and proof by induction. We give proofs that sqrt(2) is irrational and that there are infinitely many primes, among others.

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MathDoctorBob

the sad thing is you cant say the suns are better than this, bc they're not.
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Rusty Buckets

This video discusses an example of particular metric space that is complete. The completeness is proved with details provided. Such ideas are seen in branches of analysis.

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Dr Chris Tisdell

Absolute value: definition. Notion of distance. Properties of the absolute value: proofs. Triangle inequality

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Winston Ou

© 2019 How to start mobile phone business

Divide and Conquer. This is another area that is very industry-dependent, but it is highly unlikely that any company can afford to have an entire contract team devoted to managing one portfolio. More than likely, it is more realistic to divvy up the team and the contracts so that there is a leader for each relevant sphere. The entire team will obviously have to coordinate and communicate, but resources must be allocated in the most efficient manner possible. In turn, this will allow for several individuals to keep an eye on a smaller batch of contracts, thereby facilitating those periodic reviews. Outsource the Tedium to Technology.