This book was a good self learning intro to proofs for me. When I went back to school for a math degree and my undergrad analysis professor told me I am already studying like a grad student I took that as a huge compliment.
Instead of "unicorns" in your proofs, you may want to say "observed equine unicorns". Unicorn used to refer to some rhinos, and given the apparently infinite size of our universe, it is more probable that classic "equine" unicorns really exist out there than not. I cannot attest to their purpleness or math skills.
A bit late but this is one of my favorite books that I even travel with. A lot of times I have to prove stuff and if I'm stuck, I reread the relevant section, follow the advice given (reread the theorem, write the definitions, etc) and usually I end up getting the proof I need. Excellent recommendation!
Love it. I find it also related with philosophy. The reason why I related to philosophy at some point is when writing some good arguments. Even though you are not writing arguments in math (or at least not that specifically), I find it quite similar, since you need also premises to make it makes sense for writing math proofs as well. Thank you for this video
Velleman is incredibly thorough and well laid out - which can mean it takes effort to work through its content. There are several free, excellent online books, particularly Hammack Book of Proof, that complement Velleman. However, if I want a definitive answer on how to analyze a particular proposition and how to devise a strategy to tackle the proof, I turn to Velleman.
This is a book I highly recommend. It can be a grind for some of the proofs if you do not have any help from a classroom or mentor/tutor since there are no solutions. That being said there are many of the problems that are not difficult and you can build on those to help with those problems that you have difficulty with. Good suggestion. I am going through the free book recommend from another of your videos right now just to see if it brings something different. Attacking from different angles and books helps to illuminate the path to a solution. Good luck.
Statements like x^2 >= x implies x =1 always interested me when I was first studying math. Because the RHS gives a potentially larger set of values in which the LHS is true. Furthermore, the statement could be vacuous! But since the reverse direction is of course true not only is it not vacuous, but that the set of values the LHS is true is precisely given by the RHS. I think it's important to enjoy thinking deeply about things such as this if anyone wants to become a math major
I'm going through Susanna Epp's "Discrete Mathematics with Applications" and I totally see why you had it recommended as a starting book. Although I haven't gotten far into the book, I can see how the beginning chapter on logic definitely bleeds into all of mathematics!
Nice video. Based on this I ordered two books, this one and “Book of Proof” by Hammack. Hammack has odd numbered worked problems and a better rating on Amazon. At Auburn we’re we’re taught using the “Texas Method” i.e. no books, Math Prof put a few axioms on the board along with some theorems to prove for the next class. In the next class, folks were called to show their proofs and typically get shot down (except for the math brainiacs who always sat in the back rows). Then more theorems were added for the next class and so on. Problem is, no books like these too go back to and review. So you better have some serious notes to get thru the theorems on the final exam, which was 100% of the class grade, usually. The whole math department was run like this. I have 40+ quarter hours of theoretical math as an applied mathematician. Wretched! But, as a Military Operations Research Analyst it didn’t matter as you had to learn on the job as there were no classes and no textbooks anyway back in the 70s anyway. The good thing was, the Texas Method taught you to think logically, under pressure in from of a bunch of snipers…and in hindsight, I’m the better for it as it turned out.
so because we cannot concretely quantify or attach a value, action or law to basis set- unicorns this pretty much leads to infinite possibilities of statements, which we cannot prove false, then they must be true?
as a HS student who is still struggling with Algebra and Calculus, but does lots of programming which need a lot of logic Proof is kinda like Programming, but in different language. Only shapes are used
I have that book (same edition). It confused me because some of the terms did not match my bridge class. I will give it another look now. Thanks. Would a lesson on the biconditional truth table clarity your logic lesson in a more visual manner?
I know what a math statement is..but am unsure if unicorns have to be either purple or not if they don’t exist, e.i. is this really a math statement? The part about all unicorns being purple seems like it slightly depends on semantics and what’s considered the convention for how to interpret this (specifically it seems to rely on two ideas: 1: a statement either is true or false; therefore, if a statement is not true it is false, vice versa); 2: “all unicorns are purple” is a statement”. I have trouble accepting the second idea. I think I understand that if “there is a unicorn that’s not purple” is false, then the negation of said statement is true, i.e, “it’s not the case that there is a unicorn that’s not purple”. **This true statement (at least mathematically) means “for any thing that’s a unicorn, it’s purple”or “all unicorns are purple”).** THIS is where my philosophy brain has an issue, only the last premise in bold. I don’t see why this definitional convention that a statement is either true or false must extend to descriptions of things that don’t exist in reality. Cannot the claim “all unicorns are purple” be neither true or false (by a different semantic convention of what constitutes a claim)? Or perhaps, from a yet another semantic approach, can it not be considered that a descriptive statement z about something x is only true or false if x has defined parameters for what it is and z describes properties that are either consistent or inconsistent with said parameters of x? By that new convention, “all unicorns are purple” could neither be confirmed or denied, neither true or false, since unicorns are not constrained to be either purple or not purple (not by arbitrary definition nor by observable reality). In other words, unicorns don’t have parameters z that pertain to the description x of being or not being purple; therefore, “all unicorns are purple” is not true or false since there are no unicorns to which the description purple/not purple is attributable, nor is color an integral component of the definition of unicorn. Maybe I shouldn’t bring philosophy into this? Maybe the true/false dichotomy for descriptions of nonexistent things is simply more useful to us, but not logically necessary? Idk mayne!
I'm familiar with Velleman through his philosophical work, so please being philosophy into this. You seem to know what you're talking about, so for anyone that reads this and is interested further, look up Quine and truth and unicorns. Also perhaps study some of Leibniz writings on infinity and logical possibilty
By that logic, wouldn't it be the case that both statements: Unicorns are purple. and Unicorns are not purple [/Unicorns are of colors other than purple]. ...are both [vacuously] true? Honestly, the idea of vacuous truth doesn't make sense to me (does it have any practical use, or is it just a convention?). My main issue is that every statement is sort of a hypothesis about the world and its properties. And some statements have hidden premises. If someone says that Unicorns are purple, they presume that Unicorns exist. Thus it's really: Unicorns exist & All Unicorns are purple. Which ends up being false, or maybe just meaningless. Surely it's not true.
Also wouldn't the statement: Guns don't kill people; people do - be vacuously true. Obviously you don't see guns going off by themselves. It takes a person to press the trigger to release the bullet. The point is the statement is meaningless and it's a really inane argument to try to argue for no gun restrictions. The fact of the matter is guns are the most dangerous "tools" ever invented especially assault rifles. And furthermore on the logic part, you don't see cars driving themselves. (not yet) it takes a person behind the wheel to drive a car. It takes a person to bake a cake, it takes a person to write a book. So in conclusion just to reiterate, the statement that people kill people is a really dumb and vapid argument on the Republican side.
I started reading this book and was a bit disappointed with the first proof that the authors wrote in page 3 for the Conjecture 2 shown on page 2 which states Suppose is an integer larger than 1, and n is prime, then 2^n -1 is not prime. Their proof leaves much to be desired as written since it skips many steps to arrive at the line that states xy = 2^(ab) -1. This is not clearly obvious from the line above unless additional terms to the series are added so that upon cancellation one arrives at xy = 2^(ab) -1, nor is it clear why they chose this method to prove the conjecture. I submitted this conjecture to chat GTP 4 and received a much better written proof that is much clearer. I like this book but it did not get off to a good start for me. I am interested in what your thought are on this proof of conjecture 2 as given by the authors. I will read on since I think I can still learn a lot from this book. Their return to this proof on page 302 is a little better.
Hmm, they are both excellent books. They have different examples so I think it's worth having both. Note this one is a softcover but I do like it a lot.
Hey math sorcerer, I have not started mathematical logic, should I get a book that talks about it before getting this one or do they also teach it in parallel to proof writing
@@TheMathSorcerer So for a universty student of Biology, do you recommend this book? Because I don't know if my level of Maths is in part with the book
I suppose it depends on whether you define 'real' as 'existing'. If so, I think the statement would be false by definition. E(x) = ¬(x∈∅) A = {all unicorns} = ∅ ∀ x∈A : x∈A x∈∅ ¬E(x)
Another way of saying that anyone who exists in a universe without The Math Sorcerer is great at math - as long as no other universe exists. If one does, its members lose that cool quality, because they never had both it and existence.
This reminds me, and looks a lot like the frame of arguments (both inductive and deductive) taught in philosophy modules where you have; premise one, premise two and then a conclusion. (E.g. All swans are white, this is a swan, therefore this is white. If that's the case, is the mathematical notion of (vacuously) true the same as the philosophical is/exist? And I just looked up vacuously, I found that it means mindlessly or blankly etc. If that is what is meant here, is vacuously true actually just meaningless? Is it just something that we hold in our heads so we can entertain the learning point?
Yeah, "vacuous truth" seems to me to be a misnomer. The statement is either meaningless, or outright false. Every statement is a hypothesis about the nature of the world or its parts and most of the time you should be able to represent it as a conjunction of more basic statements. For instance, "The Earth is a 3rd planet from the Sun" assumes that the Sun exists, that there are some constraints as to which astronomical bodies count as planets, that Earth is a planet and that exactly two other planets are closer to the Sun than Earth (arguably you could include many many further, more basic statements there). In the same way, "Unicorns are purple" is a hypothesis that could be represented at minimum by the conjunction: "Unicorns exist" & "Unicorns are purple". The first part is false. Whether the second part is interpreted as false, [vacuously] true, or meaningless, the end result should be either false or meaningless anyway.
Here is a statement: All unicorns have 3 eyes. Is it true? If it is proved to be mathematically true by following the same procedure, that's weird and upsetting. I know the underlying text, unicorns do not exist physically, the subject of the statement is void vacuum, or its existence cannot be verified.
k o . b -ok . a si a /s/h ow% 20to%20prov e% 20i t go to that link i put them apart because over a month ago i put a link or 2 or a lot in the comments section and youtube deleted it...
This book was a good self learning intro to proofs for me. When I went back to school for a math degree and my undergrad analysis professor told me I am already studying like a grad student I took that as a huge compliment.
Kindly keep posting such type of books reviews , it helps us a lot for gaining knowledge. Great work.
Instead of "unicorns" in your proofs, you may want to say "observed equine unicorns". Unicorn used to refer to some rhinos, and given the apparently infinite size of our universe, it is more probable that classic "equine" unicorns really exist out there than not. I cannot attest to their purpleness or math skills.
Or we could say "given that no unicorns exist, all unicorns can do math"
@@iamdigory very nice
The probability of an impossibility is zero regardless of how big the sample space is. The universe is also finite not infinite.
A bit late but this is one of my favorite books that I even travel with. A lot of times I have to prove stuff and if I'm stuck, I reread the relevant section, follow the advice given (reread the theorem, write the definitions, etc) and usually I end up getting the proof I need. Excellent recommendation!
Professor I love your book recommendation videos!!!!!!!!! Those books help me a lot in my math courses and I use them as reference books.
+1 on this book. A classmate recommended it to me in graduate school. It's brilliant.
Love it. I find it also related with philosophy. The reason why I related to philosophy at some point is when writing some good arguments. Even though you are not writing arguments in math (or at least not that specifically), I find it quite similar, since you need also premises to make it makes sense for writing math proofs as well. Thank you for this video
Absolutely, maths is a branch of philosophy.
Another fantastic review video, I thoroughly enjoyed learning from you.
Velleman is incredibly thorough and well laid out - which can mean it takes effort to work through its content. There are several free, excellent online books, particularly Hammack Book of Proof, that complement Velleman. However, if I want a definitive answer on how to analyze a particular proposition and how to devise a strategy to tackle the proof, I turn to Velleman.
OMG!!!!!! I have that book, as well, Math Sorcerer!!! Thanks for yet another grand video!!! :) :) :) :)
Thanks for the review! I just loved the proof - really made my eyes light up :)
Hello MathSorcerer! Thanks to you I have a better grasp on proof by inductions and logic!
I'm from brazil, your videos are very good! Thanks! I study phd in economics. This book help me understand and make proofs!
7:41 there *are* unicorns, but we call them rinos because Hollywood imposes unreallistic beauty standards on unicorns #realunicornshavecurves.
I have the second edition of that book which does contain solutions to selected excersizes in appendix 1.
i am so proud that i understood the title as "vacuously true"
This is a book I highly recommend. It can be a grind for some of the proofs if you do not have any help from a classroom or mentor/tutor since there are no solutions. That being said there are many of the problems that are not difficult and you can build on those to help with those problems that you have difficulty with. Good suggestion. I am going through the free book recommend from another of your videos right now just to see if it brings something different. Attacking from different angles and books helps to illuminate the path to a solution. Good luck.
Where would we be without the good ol’ null set? Never has emptiness been so important!
Statements like x^2 >= x implies x =1 always interested me when I was first studying math. Because the RHS gives a potentially larger set of values in which the LHS is true. Furthermore, the statement could be vacuous! But since the reverse direction is of course true not only is it not vacuous, but that the set of values the LHS is true is precisely given by the RHS.
I think it's important to enjoy thinking deeply about things such as this if anyone wants to become a math major
Also the reverse implication comes for free in algebra because algebraic steps tend to be if and only if lol
So, does that mean that "all unicorns can't do math" or "all unicorns are not purple" is also true mathematically?
@Gustav Interesting. Is your sentence about elements a statement?
@Gustav I mean are you suggesting to check if it's true or not?
I'm going through Susanna Epp's "Discrete Mathematics with Applications" and I totally see why you had it recommended as a starting book. Although I haven't gotten far into the book, I can see how the beginning chapter on logic definitely bleeds into all of mathematics!
that book is nice, i'm reading it
My intro to proofs book!
Does anyone know of a dictionary of all mathematical symbols?
Nice video. Based on this I ordered two books, this one and “Book of Proof” by Hammack. Hammack has odd numbered worked problems and a better rating on Amazon. At Auburn we’re we’re taught using the “Texas Method” i.e. no books, Math Prof put a few axioms on the board along with some theorems to prove for the next class. In the next class, folks were called to show their proofs and typically get shot down (except for the math brainiacs who always sat in the back rows). Then more theorems were added for the next class and so on. Problem is, no books like these too go back to and review. So you better have some serious notes to get thru the theorems on the final exam, which was 100% of the class grade, usually. The whole math department was run like this. I have 40+ quarter hours of theoretical math as an applied mathematician. Wretched! But, as a Military Operations Research Analyst it didn’t matter as you had to learn on the job as there were no classes and no textbooks anyway back in the 70s anyway. The good thing was, the Texas Method taught you to think logically, under pressure in from of a bunch of snipers…and in hindsight, I’m the better for it as it turned out.
Amazing as always.
Thanks
There is a sequel to How to Prove it. Don't know the title, but it does not play on the title from Polya's book.
Have you covered Polya's How to Solve It?
Yup
I bought this book and started to read it. My brain began to hurt. I'm now watching this video until my brain cools down so I can get back to it.
Great video.
so because we cannot concretely quantify or attach a value, action or law to basis set- unicorns this pretty much leads to infinite possibilities of statements, which we cannot prove false, then they must be true?
I've been saying that to my friends for years.
Do you like this book better than Jay Cumming's "proofs"?
You made me buy another book! XD
as a HS student who is still struggling with Algebra and Calculus, but does lots of programming which need a lot of logic
Proof is kinda like Programming, but in different language. Only shapes are used
I have that book (same edition). It confused me because some of the terms did not match my bridge class. I will give it another look now. Thanks.
Would a lesson on the biconditional truth table clarity your logic lesson in a more visual manner?
6:40 Time to get out the sledgehammer, *cracksnuckles* whips out the empty set, Time to get serious
Yep, I love this book too
I know what a math statement is..but am unsure if unicorns have to be either purple or not if they don’t exist, e.i. is this really a math statement? The part about all unicorns being purple seems like it slightly depends on semantics and what’s considered the convention for how to interpret this (specifically it seems to rely on two ideas: 1: a statement either is true or false; therefore, if a statement is not true it is false, vice versa); 2: “all unicorns are purple” is a statement”. I have trouble accepting the second idea. I think I understand that if “there is a unicorn that’s not purple” is false, then the negation of said statement is true, i.e, “it’s not the case that there is a unicorn that’s not purple”. **This true statement (at least mathematically) means “for any thing that’s a unicorn, it’s purple”or “all unicorns are purple”).** THIS is where my philosophy brain has an issue, only the last premise in bold. I don’t see why this definitional convention that a statement is either true or false must extend to descriptions of things that don’t exist in reality. Cannot the claim “all unicorns are purple” be neither true or false (by a different semantic convention of what constitutes a claim)? Or perhaps, from a yet another semantic approach, can it not be considered that a descriptive statement z about something x is only true or false if x has defined parameters for what it is and z describes properties that are either consistent or inconsistent with said parameters of x? By that new convention, “all unicorns are purple” could neither be confirmed or denied, neither true or false, since unicorns are not constrained to be either purple or not purple (not by arbitrary definition nor by observable reality). In other words, unicorns don’t have parameters z that pertain to the description x of being or not being purple; therefore, “all unicorns are purple” is not true or false since there are no unicorns to which the description purple/not purple is attributable, nor is color an integral component of the definition of unicorn. Maybe I shouldn’t bring philosophy into this? Maybe the true/false dichotomy for descriptions of nonexistent things is simply more useful to us, but not logically necessary? Idk mayne!
I'm familiar with Velleman through his philosophical work, so please being philosophy into this. You seem to know what you're talking about, so for anyone that reads this and is interested further, look up Quine and truth and unicorns. Also perhaps study some of Leibniz writings on infinity and logical possibilty
@@quintinpace2627 I appreciate it
The empty set - where dreams are shattered
How about a video on a couple Logic books to pick from to help us do that logic part you referred to??
By that logic, wouldn't it be the case that both statements:
Unicorns are purple.
and
Unicorns are not purple [/Unicorns are of colors other than purple].
...are both [vacuously] true?
Honestly, the idea of vacuous truth doesn't make sense to me (does it have any practical use, or is it just a convention?).
My main issue is that every statement is sort of a hypothesis about the world and its properties. And some statements have hidden premises.
If someone says that Unicorns are purple, they presume that Unicorns exist. Thus it's really: Unicorns exist & All Unicorns are purple. Which ends up being false, or maybe just meaningless. Surely it's not true.
Also wouldn't the statement: Guns don't kill people; people do - be vacuously true. Obviously you don't see guns going off by themselves. It takes a person to press the trigger to release the bullet. The point is the statement is meaningless and it's a really inane argument to try to argue for no gun restrictions. The fact of the matter is guns are the most dangerous "tools" ever invented especially assault rifles. And furthermore on the logic part, you don't see cars driving themselves. (not yet) it takes a person behind the wheel to drive a car. It takes a person to bake a cake, it takes a person to write a book. So in conclusion just to reiterate, the statement that people kill people is a really dumb and vapid argument on the Republican side.
What fun! Thank-you.
I started reading this book and was a bit disappointed with the first proof that the authors wrote in page 3 for the Conjecture 2 shown on page 2 which states Suppose is an integer larger than 1, and n is prime, then 2^n -1 is not prime. Their proof leaves much to be desired as written since it skips many steps to arrive at the line that states xy = 2^(ab) -1. This is not clearly obvious from the line above unless additional terms to the series are added so that upon cancellation one arrives at xy = 2^(ab) -1, nor is it clear why they chose this method to prove the conjecture. I submitted this conjecture to chat GTP 4 and received a much better written proof that is much clearer. I like this book but it did not get off to a good start for me. I am interested in what your thought are on this proof of conjecture 2 as given by the authors. I will read on since I think I can still learn a lot from this book.
Their return to this proof on page 302 is a little better.
great video👍
new edition has chapter on number theory also
Sir can you please make a video on doing fast multiplication division addition and subtraction quickly, please.
P(x) is an open sentence, not a statement since we don't know the truth value of x. Me thinks.
Nice 😊
Hey MS,
How is this book compared to the book by Gary Chartrand, Albert D. Polimeni & Ping Zhang
Hmm, they are both excellent books. They have different examples so I think it's worth having both. Note this one is a softcover but I do like it a lot.
@@TheMathSorcerer
Thanks for replying
Do you have to mathematically prove Unicorns don't exist for the proof to 'be valid'?:)
If you go to Chegg or quizlet, you might be able to find solutions, but I couldn't guarantee if they are correct.
"I've also done several of the exercises, which I didn't do in bed."
Hahaha! Thanks for that! That is very important advice.
Rofl
Is this book better than the book of proof?
I think so yes
Hey math sorcerer, I have not started mathematical logic, should I get a book that talks about it before getting this one or do they also teach it in parallel to proof writing
The book in the video teaches it all👍 link in the description 👍👍
@@TheMathSorcerer So for a universty student of Biology, do you recommend this book? Because I don't know if my level of Maths is in part with the book
Yes definitely. This is a wonderful book and one of the best!!
Also it’s not super expensive.
@@TheMathSorcerer Okey, thank you very much
Any good specific math logic books and courses for sentential logic, predicate logic and graph theory ?
Is this book going to feel boring if you already thoroughly studied through Susanna S. Epp's Discrete Mathematics with Applications book?
Nice pencil!
BTW I think there are selected solutions on the publisher web page. Not 100% certain…
hello Sir Alex Ferguson...have a nice day
If i got this right, ¿would that mean that:
"Unicorns are not real" is a true statement
"ALL unicorns are real" is also a (vacuously) true statement
?
I suppose it depends on whether you define 'real' as 'existing'. If so, I think the statement would be false by definition.
E(x) = ¬(x∈∅)
A = {all unicorns} = ∅
∀ x∈A : x∈A x∈∅ ¬E(x)
How many copy of book u have??
just 1 copy of this book
@@TheMathSorcerer hahahah .. I am asking about collection of all of ur book.
Another way of saying that anyone who exists in a universe without The Math Sorcerer is great at math - as long as no other universe exists. If one does, its members lose that cool quality, because they never had both it and existence.
I know a few unicorns. Luckily, they all can do math :D
Don’t forget your Halmos tombstone!
The downside of this proof is that unicorns did exist at one time but probably not on this planet.
This reminds me, and looks a lot like the frame of arguments (both inductive and deductive) taught in philosophy modules where you have; premise one, premise two and then a conclusion. (E.g. All swans are white, this is a swan, therefore this is white.
If that's the case, is the mathematical notion of (vacuously) true the same as the philosophical is/exist?
And I just looked up vacuously, I found that it means mindlessly or blankly etc. If that is what is meant here, is vacuously true actually just meaningless? Is it just something that we hold in our heads so we can entertain the learning point?
Yeah, "vacuous truth" seems to me to be a misnomer.
The statement is either meaningless, or outright false.
Every statement is a hypothesis about the nature of the world or its parts and most of the time you should be able to represent it as a conjunction of more basic statements.
For instance, "The Earth is a 3rd planet from the Sun" assumes that the Sun exists, that there are some constraints as to which astronomical bodies count as planets, that Earth is a planet and that exactly two other planets are closer to the Sun than Earth (arguably you could include many many further, more basic statements there).
In the same way, "Unicorns are purple" is a hypothesis that could be represented at minimum by the conjunction:
"Unicorns exist" & "Unicorns are purple".
The first part is false. Whether the second part is interpreted as false, [vacuously] true, or meaningless, the end result should be either false or meaningless anyway.
Can you do the induction for saying any animal can do maths? haha😂😂😄😄😁😁
100th like. There is a 2nd edition to this excellent book.
Am the best ✌️✌️✌️✌️✌️✌️✊✊✊✊
❤️❤️❤️❤️❤️
I love this book!
Am always first
Not all Mathematicians can do Unicorn.
wait that doesn't sound right...
Here is a statement: All unicorns have 3 eyes. Is it true?
If it is proved to be mathematically true by following the same procedure, that's weird and upsetting.
I know the underlying text, unicorns do not exist physically, the subject of the statement is void vacuum, or its existence cannot be verified.
Yeah it's true because they don't exist in the first place. Anything you can't prove to be false is considered true. There's nothing in between.
@@ILoveMaths07 *doxastic logic starts sweating in the corner
All infinites are odd
All infinites are even
👍👍👍👍👍👍👍
I dont like yo brag but I created smother working idea in cal 3
It's a fairly good book on mathematical proofs, but fails completely for logical proofs.
Every man on Mars is 900 years old. It is true.
If you are reading this, then you are a unicorn.
To show the statement is false give an example of a unicorn that cannot do math.
Non-purple unicorns do not exist!!
Hahahaha
Unicorns are narwhals so...
:000
Your conjecture All {Unicorns} thingy,
∀ {🦄} = ✍⟺ ∃🦄∈ 🦹♂˄ Subd
k o . b -ok . a si a /s/h ow% 20to%20prov e% 20i t
go to that link i put them apart because over a month ago i put a link or 2 or a lot in the comments section and youtube deleted it...