4 comments about Problems in Real Analysis: A Workbook with Solutions.

1. Before buying this book, I was failing Real Analysis. Now I have a prayer of passing. Thank goodness I found it in time. For it to be of use, you need to buy the companion book "Principles of Real Analysis", same authors. On the down side, it doesn't have an index, but overall, well worth the money. Besides, it is the only source I've found of worked-out Real Analyis problems outside of borrowing from other students who have already taken the course.



2. Great guide, must have for anyone taking Real Analysis.



3. This book rocks! It covers practically all the major topics of an introductory course in graduate Real Analysis. Excellent solutions that aid in the understanding of the material. This book's worth is immeasurable, or should I say, non-measurable.



4. There is no problem solutions for Real Analysis texts available in U.S. since most of the teachers believe that Math students in grad level should be more creative. However, not all students in Real Analysis are potential mathmatician. If they lost in class and must study by theirselves, they may feel frustrated missing all the stuff contained in problems. The material in "Principles of Real Analysis" may not superior much than other famous text(like Royden, I think Royden is clear enough but too much mysterious things lie in problems.). But use it with this workbook, you will find much comfortable in self study. It helps a lot not only in my homework assignment, but also in my understanding.

5 comments about Calculus Demystified : A Self Teaching Guide (Demystified).

1. I bought this book, along w/ several others, as prep for starting a masters program. I had completed Calc 1 - 3 as part of my undergrad many years before, but had been away for so long I knew I would need a refresher on Pre-Calc as well as early Calc.
   
   I wanted the book to be a lead-in to re-taking Calc 1, so I was looking for some quick review of Pre-Calc topics, then the meat of the book to be Calc 1-2 topics.
   
   I never got past the first chapter with this book--
   * Reference/Editors notes appear to be left in the text in some locations -- These were things like "If you need more help see [SCH1]", of course "[SCH1]" means nothing to me (the reader), because it does not appear to be a chapter marker or even a reference to another book.
   
   * I found too many of the problems are presented w/o a step-by-step demonstration (or even a solution) -- How can I tell if I am successfully working a problem if I do not know the end result?
   
   * Several of the problems they provide w/o solutions contain material that was not in the chapter, yet is required to begin the problem.
   
   In general the book did not help me at all. I gave it two days, assuming the first day was just general frustration, when the second day resulted in the same feeling I moved on to another book.
   
   I recommend a different book--"Calculus For Dummies"



2. i personally bought this product for re studying calc that i had already learned back in high school. ITS GREAT!!!!



3. Whatevery your plan is, do not rely on this book. If you are reviewing the subject, this book will make you feel stupid. If it is your first time learning Calc., it will stop you from persuing mathematics.
   The biggest problem is that it does not really teach you the material. All it does is give you an idea of how it works, and then gives you hopelessly complicated examples (which will have you thinking 'I will never learn calculus!'.)
   If you are into 'the hardest possible way to learn something' only THEN is this book for you.



4. this is a cheap book that lacks many topics covered in my college calculus class.



5. I have to agree with most of the negative comments posted here. I took several courses in calculus in college but wanted to relearn it quickly. Instead of dragging out my old texts, I bought of copy of this book. Big mistake. The volume is written for mathematicians. There are few practice problems, and the ones that are included do not have solutions! The chapter exams consist of extremely difficult problems, making the use of this text discouraging. I ended up buying Calculus for Dummies and using my old texts to get problem sets.
   
   I would add that I also bought the 'Demystified' volumes on linear algebra and differential equations, and both of them were quite good. The calculus volume seems to be a dud, though.

5 comments about Schaum's Outline of Graph Theory: Including Hundreds of Solved Problems.

1. In this book one can find a practical survey of both principles and practice of graph theory, with great coverage of the subject. The outhor provides a lots of solved problems, with losts of theory proofs and all with great clarity and common reasoning. The outhor gets you enter the subject step by step from the easy problems to the hardest with great skill. Also the algorithms on graphs presented in this book, and in general the algorithmic approach of this book are presented most clearly. You wouldn't leave this book until you'l finish read it and understand graph theory. Finally you would fill that at least on one branch of mathematics you are well sitted.



2. In general, the book not requires study in advance, but it is better for reference. I'm a software engineer and the book's treatment of "Shortest Path" and "Connectivity" problems is very usefull. Good for fast remember of the subject.



3. This book was an absolute hell to contend with. I've taken two courses in Graph Theory, using Robin J. Wilson's Introduction to Graph Theory and this cheap broadsheet, respectively. Wilson's book is the one to use! It's extremely well-written, even fun to read--the reviews on Amazon will bear that out.

   In the second graph theory course that I took (to refresh and refine my understanding), the professor chose the Schaum text solely for its low cost--he thought he was doing the students a service. Hardly.

   No thought whatsoever has been put into the readability of this book. The tiny dark-grey font on light-grey paper is a simple enough design flub that makes reading past even two or three pages at a time almost unbearable. Defining terms is seen as a chore to be compacted--a single page at the beginning of each chapter might try to define 10-15 terms, just to get them out of the way. It becomes a mess of bold print that the reader is forced to continually return to because the definitions come with no context nor examples by which to remember them. In the end, the reader realizes that 2/3 of the book is just list after list of badly-worded questions following under-scripted lessons.

   Look, it's not even worth writing any more about, the text frustrates me so much. There's only two other reviews on this page, and I'd place money on them being written by the author himself. Save yourself the $$$ and the hassle, and just go buy Wilson's book. Trust me.




4. I have bought and used many Schaum's outlines on various subjects in math and science, and I would say that this outline on graph theory is one of the worst. Most Schaum's outlines give you the theory in small doses, with plenty of diagrams to explain the concepts. This outline reads more like one of the textbooks on the subject, however. Theorems and their illustrations are poorly presented, and the author could not have made the subject matter drier and more unappealing if he had tried. You might be able to get something out of it if you are a student of pure mathematics, but you will definitely be disappointed in this book if you are a computer science student. If you are already using a bad textbook for a class in graph theory, this book will only add to your collection of bad unreadable texts on the subject. For computer science students, I suggest that you check out the chapters on graph algorithms in Introduction to Algorithms by Cormen et al. That book has pseudocode, explanations, and diagrams to help you work out implementations of various graphing algorithms.



5. This book is wonderful in my eyes. However, I do not recommend most Schaum's Outlines as textbooks, but as supplements to texts. They just contain too much. This book is good reference to have if you're doing a course in graph theory or if your work involves graph theory. I highly recommend it for reference use.

5 comments about Calculus: One and Several Variables, Ninth Edition.

1. I used this book for calc 2 and am currently using it for calc 3 (I have a test tomorrow....AHHHHHH). I don't really like it. It doesn't explain a lot of stuff in a way easy to understand. Also there aren't good examples for the problems, so you get to homework and you have no idea how to actually do it because there isn't an example of it. But it's not the worst book ever. If you think hard enough you can usually follow what the text is attempting to explain to you. The Student Solutions Manual is helpful, but can lead to more confusion when it's wrong or it solves something in some obscure totally weird way.



2. If you dread taking calculus and aren't into math theory, I wouldn't recommend using this book. It'll only increase your headaches. The examples are sometimes too simple for the homework problems and will require an additional supplement. The chapters are horribly structured. A new chapter usually begins directly underneath exercises--a definite eye sore, given all the theorems shown. I, however, think the chapter on "Limits" is by far the best section. If the book implemented this method of teaching throughout the whole book, I'd give the book a better rating.
   
   A college calculus book I can stand behind is "Calculus" by Smith and Minton. They work out several examples in simple terms to help you understand Calculus without all the frustration. I wish I would have discovered it earlier during my course.



3. The important theorems are proved, and most everything is worked out in a very straightforward manner. For those who want theory, this text has it. For those who don't, there are a great many examples that guide the student through nearly any problem. The exercises are challenging and relevant.



4. Having used this book throughout my undergraduate education in calculus (Calc 1-3), I can attest to the effectiveness of this edition. What I truly love about this edition is the effort put forth to actually explain why, and not just give a "that's the way it is" analysis of concepts. After meeting other people exposed to calculus in high school where you're expected just to take concepts like the connection between the definite and indefinite integral at face value, it was amazing to actually come across a calc book that takes time to explain things logically, which is kind of the whole point.
   Word to the wise, this isn't a calc for business or calc for life sciences version. This is calculus, pure, pulling no punches. I'm giving it four stars because I think there could have been better organization concerning the proofs for some of the ideas and concepts presented.



5. This is an excellent textbook as a classroom text or for self-study. I
   have read other reviews where the reviewer complains about too much theory. This is nonsense. If you only want formulas then take only that
   from the book. If you want more it is there. This book is an excellent
   precursor to analysis and differential equations. The authors don't baby
   the reader but you better come well versed in precalculus. Please don't
   try to study calculus if you can't do inequalities, logs, etc. and then
   blame the author and your instructors for your failures.

5 comments about Functional Analysis.

1. De los excelentes textos en Análisis Funcional que existen en el mercado, éste es de los mejores. Tiene una excelente presentación de la Teoría de Distribuciones, en los capítulos 6, 7 y 8. La teoría espectral como se trata aca es magnifica. Tambien tiene un desarrollo muy completo sobre espacios vectoriales topológicos. Termina con una reseña bibliográfica muy completa.



2. No other book covers the elements of distributions and the fourier transform quite like Rudin's Functional Analysis. This is a must for every budding PDE-er!



3. "Modern analysis" used to be a popular name for the subject of this lovely book. It is as important as ever, but perhaps less "modern". The subject of functional analysis, while fundamental and central in the landscape of mathematics, really started with seminal theorems due to Banach, Hilbert, von Neumann, Herglotz, Hausdorff, Friedrichs, Steinhouse,...and many other of, the perhaps less well known, founding fathers, in Central Europe (at the time), in the period between the two World Wars. In the beginning it generated awe in its ability
   to provide elegant proofs of classical theorems that otherwise were thought to be both technical and difficult. The beautiful idea that makes it all clear as daylight: Wiener's theorem on absolutely convergent(AC) Fourier series of 1/f if you can divide, and if f has the AC Fourier series, is a case in point. The new subject gained from there because of its many sucess stories,- in proving new theorems, in unifying old ones, in offering a framework for quantum theory, for dynamical systems, and for partial differential equations. And offering a language that facilitated interdisiplinary work in science! The Journal of Functional Analysis, starting in the 1960ties, broadened the subject, reaching almost all branches of science, and finding functional analytic flavor in theories surprisingly far from the original roots of the subject. The topics in Rudin's book are inspired by harmonic analysis. The later part offers one of the most elegant compact treatment of the theory of operators in Hilbert space, I can think of. Its approach to unbounded operators is lovely.



4. Hardly can I find words to highlight the goodness of this book. As mentioned by other readers ,it provides elegant, direct and powerfool proofs of the three theorems which constitute the cornserstones of functional analysis (Hanh-Banach, Banach-Steinhaus and Open mapping). These theorems are, in addition, studied in their most general context, namely topological vector spaces.
   
   Specially appealing is its treatment of distributions' theory. It is, as far as I know, the only text which start by defining the rigurous topology on the set of test functions and then obtains the convergence and continuity of functionals (distributions) in terms of this topolgy, which is, indeed, the only way to present and gain insight into these concepts and to reach some results such as completness. In doing otherwise one risk definitions can emerge as artificial and rather arbitrary.
   
   It is, without any doubt, a must have book for those with interest in pure mathematics as well as for those who, eventually, realize that the only way to dominate their area is saling through mathematics.



5. I enjoy perusing Rudin's "Functional Analysis" at this stage in my life. It is fairly nice tome for functional analysis, and its general treatment of topological vector spaces (as opposed to the standard Banach space examples studied in a typical functional analysis class) is now well-received.
   
   However, as a student, I was put off by this book. At times, I found it difficult to tie the theory present to the basic examples which were relevant at the time (such as L^{p} spaces). For a first time learner, I would suggest the book of Kolmogorov and Fomin (which is a Dover book, by the way), and would wait until later for this book.

5 comments about Calculus (College Review Series).

1. This text is a nice balance between a traditional calculus text and the smarmy calculus by cartoon type books, and it is more math book like than 'A Tour of the Calculus' by David Berlinski in that you get examples and practice problems.
   The explanations are written in a relaxed, literate, and very readable style, without being patronizing or silly.
   Enough examples and practice problems are provided to get the key points pounded into your head. The examples are worked through step by step with fairly clear explanations.
   Be warned. This is definitely a review book or to be used in conjunction with a traditional class and text. It's a very quick pass over the material. It provides some minimal algebra review, but if you are rusty you will want some practice since it assumes you can handle rational expressions and exponents.
   I worked through it a chapter at a time over an otherwise lazy week.



2. Professor Gootman is the master of moving from practical everyday arithmetic to higher layers of algebraic abstraction. In Calculus, I too memorized the formulas, rules, etc. and did fine but never really fully understood the purpose of it all. To start with the simple notion of s being a 'position' of an object (ball thrown up in space) and t being time and answering the 'instantaneous rate of change' / feet per second for s(t) was such a refreshing explanation to see. Moving carefully into the next layer(s) of abstraction ( f(x) dy/dx,... ) is his forte. He helped me feel more confident knowing that even with subjects such as abstract algebra and number theory, remembering to try and move carefully 'up' the levels of abstraction will assuredly alleviate pain and frustration.



3. Great book it helped me pass Calc in high school and helped me get an A in college. Highly recommended. Easy to read and understand



4. This book is well written and the author explains the material in an easy to understand manner. I haven't had Calculus for over 10 years and a lot of the material is coming back to me because of the author's style of presenting it. I whole-heartedly recommend this review for those who have been away from Calculus for a number of years.



5. This book doesn't cover all there is to cover in Calculus 1. It can't, it's too tiny. But what it does do it take you through the logic of increasingly abstract concepts. I found this enormously helpful beyond just helping me understand the concepts (though it did so admirably). I found that this careful progression helped me formalize my own thought process, helped me get more logical.
   
   This won't cover everything you need to know, but I'm a big advocate of the more you know, the better, and this book will help you fill in some gaps.

5 comments about Understanding Analysis.

1. Once in a while, a book comes along that is so wonderfully written, the reader reflexively searches for other books by its author. Understanding Analysis is a prime example of this rare breed (Unfortunately, this is Abbott's only book as far as I know: write more!).
   
   Undergraduates often begin analysis courses with dread and finish in a state of utter confusion,knowing the definitions of key phrases, and sometimes even being able to supply proofs for some elementary results, but having no intution as to why the main theorems are pertinent.
   
   But it does not have to be so. 'Understanding Analysis' has the distinction of being so readable, it is sometimes difficult to pry oneself away from its pages and attempt the exercises. On multiple occasions I found myself skimming through the book and reading the various 'special topics' (e.g. Cantor Sets, Integration, Fourier Series) interspersed throughout the book to pique the readers' interest. But most importantly, a reader will come away with an understanding of many theorems in analysis. He or she will begin to develop a vocabulary of results that make sense both mathematically and intuitively, be able to use the results to complete the exercises (which are by no means simple 'plug-and-chug' problems), and be excellently prepared for study at a more advanced level.
   
   Bottom line: Abbott's book may not be encyclopedaic in content, but it, without a doubt covers a sufficient amount of material to warrant its use for a one-semester course in analysis. My only concern is that after such a fantasticly lucid treatment, students may have difficulty adapting to the vast selection of more advanced, less pedagogical texts available. I sincerely hope Abbott writes a sequel.



2. When I started reading analysis, I was unfortunately asked to start with Rudin's book. But that book was totally inacessible probably because I come from engineering background. In search of more readable books, I started reading Apostol. It was readable but I wasn't enjoying the subject. Recently, I came across Abbott's book and was totally blow away. It is simply amazing. It makes analysis enjoyable and at the same time you learn a lot.



3. If you're attempting to learn real analysis in one dimension, Abbott's Understanding Analysis is a great place to start. It is everything that a math textbook used for instruction should be: it has clean, concise prose, it assumes modest jumps in understanding, and it includes a good selection of exercises. Additionally, Abbott's book maintains a conversational tone without watering down the formality at the center of the mathematics while managing to convey the feeling of seeing "the big picture". Yes, there are more complete treatments (Rudin, Bartle, Browder, etc), but none of them are nearly as accessible, and frankly they aren't as good at providing an introduction to the subject.
   
   This last statement may cause cries of anguish from mathies everywhere, as I've just suggested that there are some ways in which this book is better than Rudin's Principles of Mathematical Analysis. Rudin's texts (and most other upper division and graduate math texts that I've read) seem to fall into the same pedagogical trap: they assume that the student is already familiar with the material, but they may need a quick reminder of the particulars. This is, of course, not generally the case, so the student is left to fill in whatever gaps exist, hopefully with the aid of an instructor. Indeed, there is a sort of book for which this strategy is ideal: a reference. For this use, Rudin is spectacular. For actually learning the material for the first time, it is useful to have a bit of guidance, a bit of context, and a bit of direction. It is as if many math authors have forgotten a time where they didn't thoroughly understand the material, or worse, that they have somehow conflated the pain that they experienced as students while trudging through poorly realized texts with learning the material! Abbott does not fall into this trap, and for this, he deserves more praise than I can manage. The quality of the exposition in this book has re-awakened my dissatisfaction with most other math texts.
   
   The only negative comments that I can make about this book come as a direct consequence of the material that Abbott chose not to cover. The chapters are as follows: the real numbers, sequences and series, basic topology on the reals, functional limits and continuity, the derivative, sequences and series of functions, the Riemann integral and additional topics, which include the generalized Riemann integral (a.k.a the gauge integral), metric spaces and the Baire category theorem, Fourier series and a construction of the reals from the rationals. All of these topics are done with respect to the real line, and there is no move toward generalizing the results to multiple dimensions.
   
   I desperately want to see this book in general use, but for this to happen I think that it needs to cover sufficient material for a year long sequence. If Abbott were to include material on real analysis in n-dimensions (including vector valued functions), more information on metric spaces, and an introduction to function spaces, that should do it.
   
   To summarize: if you're trying to learn the material presented in this book, buy it, but beware: the quality of the exposition of this book will spoil you and make you dissatisfied with other texts.



4. This is not a bad book. However, I dont 't understand how some reviewers claim that this book is ideal for the beginning student. Yes some things are explained very clear, however the reader should be aware that this book contains a lot of gaps left as an exercise for the reader. And I think that most beginning undergraduate students will not be able to complete these gaps without the help of a good teacher.
   If you want a very good book for beginning abstract analysis, I would rather recommend "Real Mathematical Analysis" of "Charles Chapman Pugh". Pugh's book is excellent: it is very clear written, motivates the reader by providing the necessary background details that puts everything in the right context (like Abbott does also in an excellent way) but full proofs are present. In Pugh's book, occasionally some proofs contain little gaps left as exercise, but if they do, these gaps are more fair than the gaps in Abbott' s book, if you understand the material you should be able to solve them without a guiding teacher. And indeed Pugh also has very challenging exercises, but het does not mix them up with the theory, wich is more fair to the reader.
   An additional plus, in contrast wih Abbott, is that Pugh's book contains more abstract material and is fully n-demensional.



5. I am a beginner in learning Analysis and I took Calculus I and II and got A's. However, I feel that there are not enough examples and proofs for theorems in this book. A lot of proofs are left for exercises. I cannot imagine and don't understand how beginning students can solve the exercise problems with so little examples!!????? I am taking this course right now and I am studying so hard---read books, notes and do whatever I can. However, it takes me more than 20 hours to do homework each week for this single course!! I must have a math tutor to help me solve homework problems. Each time, even with the tutor's help, it takes me a horrible long time to write the proofs down. For beginning students, without examples, how can they create concise proofs by themselves??? I am at least a good student and hard-working. I only recommend this book for complementary reading!! Right now, tonight, I have to stay up for this week's homework again! I am already studying all the time. What happened? Is this book for beginners?

5 comments about Quick Calculus: A Self-Teaching Guide, 2nd Edition.

1. I used this book to prepare for some graduate work in geosciences. I found that it was an excellent text for getting up to speed and comfortable with single variable calculus. However, the coverage of multivariable calculus is very basic. There is no vector calculus, grad, curl, etc. So this will get you started, but for most applications you will still have a long way to go.



2. This book has helped me review my calculus that has long been forgotten. It starts with review of algebra, trigonometry, and pre-calc material. Then dives into single variable derivatives and integrals. I don't think there is any multi-variable calculus.



3. This is a great book for refreshing your knowledge of basic calc. It is very fast to go over. It teaches by using problem examples with increasing difficulty. There is very little repetition, and each concept or type of problem is only in the book once and possibly twice at most. That said, I would not recomend this for someone that has never taken any calculus in their life.
   
   Otherwise, I covered log functions, and derivatives in only a couple hours with great comprehension!



4. Like many of the other reviewers, this book was invaluable for me at an earlier time. It teaches in small, digestable bites, and provides reinforcement of what it teaches. A person MIGHT do as well with a conventional problem book, but only with a great deal of discipline.



5. After using this book for a tuorial i would not recomend it to anyone. There is no substitution for a text that explains the resons you aer doing what you are doing. This book walks you through problems as you progress. If you dont understand something you have nothing other than the problem you dont understand to work with. DONT USE THIS BOOK!

5 comments about Introduction to Real Analysis, 3rd Edition.

1. Honestly this is a 4 star book, but like many of the advanced math textbooks the average score is too low, because of reviewers who clearly did not understand what they were getting into.
   
   Probably the best piece of advice with regards to advanced math books like this is given in the "Preface to the Student" in Sheldon Axler's Linear Algebra Done Right, he states: "You cannot expect to read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast."
   
   If you study from this book from that standpoint, you will get a lot out of it. But its a serious commitment.



2. I'm using this book for my real analysis course at University of Illinois and I love it. Most readers seem to be upset that some of the material isn't presented as easily as it could be, but this book is an introduction to real analysis, not to math. This is not a good book for people who have never written or read formal proofs or who are not familiar with concepts like the triangle inequality. This is a good book if you are familiar with formal mathematics and have interest in real analysis.



3. In Analysis I, we used the first 6 chapters of this book and now in Analysis II, we're covering most of the rest. This book is quite good. When I use this book, I often sit in front of a computer so that I can look up anything I don't understand. Basically, I study Analysis all from this book, Wikipedia, and random sites. I do agree that if you don't have access to full solutions to some problems at first, it's not easy to imitate the proofs in the book.
   
   One thing to point out is the exercises are quite crucial. If you don't do most, you will miss many subtleties that are not explicitly mentioned. For example, a function that is continuous except for countably many (maybe infinite) discontinuities is integrable. Another thing to note is for Chapter 7 which is more succint and to the point in 2nd ed. than in the 3rd.
   
   If you want solutions, with a bit of search on google you can find quite many. The key is to search for Universities that use this book and check out the course webpage.
   
   In Analysis III and IV, we use Rudin.



4. I have read this whole book for a Phd qualifying exam, mastering all the proofs and solving almost all the excercises, excep for the sections on numerical methods. I can say that this book is a masterpiece.
   The proofs are clear and easy to follow, and the book flowes smoothly. I can say that it is a classic in its filed as Royden's Real Analysis (3rd Edition), Churchill's Complex Variables and Applications, Fraleigh's First Course in Abstract Algebra, A (7th Edition) and so on.



5. Like some others, I really disliked Real Analysis at first b/c the proofs were so much more complex than anything else I had seen. I struggled, ordered other analysis books to help me, only to find that this one really is good! You do need a great instructor to go with this book or you may be lost. That said, the appendices are fantastic and the authors give "hints" (and some answers) to selected problems. The proofs themselves are terse, so without an instructor who understands the gaps, you may not connect the steps solo. Good text which is now part of my math library.

5 comments about Thomas' Calculus (10th Edition).

1. My boyfriend took two semesters of Calc with this book and was able to essentially self teach himself all of it. The examples are well explained and the problems reflect the section material very well.



2. I just completed the Calc series using Thomas Calculus. In Calc I, the instructor commented that we would use this text and that it wasn't the best or the worst (a VERY safe claim to make). I would agee with that.
   
   I found the instructional text adequate but we experienced several cases where the solution to end-of-section problems was not the technique being covered in the section.
   
   I used the softbound vol 1 & 2. You can save some money if you are taking a semester or two of calculus, but if you intend to cover Calc III & IV, I recommend buying the hard bound.



3. Examples are too hard to follow which makes this book difficult to read. I spent the whole semester crashing my head on the wall because of this book.
   
   Better get Calculus by James Stewart



4. Great for students in engineering or science--but not for mathematics majors who will go on to take analysis and algebra. They deserve a treatment by Spivak or Apostol. There are some interesting problems that follow the end of each chapter. Overall, it's not that interesting, but not that bad in terms of material presentation.



5. I liked how this book gave many detailed graphs to help explain the concepts. My only complaint so far is how it vaguely explained to apply the Divergence theorem and then gave a pretty ambiguous picture that was pretty worthless to understanding what exactly it applied to.