Proofs

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/sci/ has got no clue about proofs.

How do I prove that?

>>7683708
Interesting book. I have a question but I won't make a new thread about it because it is probably irrelevant.

As someone who is doing proofs in school in things like linear algebra, I decided to check the book out and see if I'm not retarded. I skipped the walls of text and went immediately to the examples.

See this:
>Example 3.1.3. Suppose a, b, and c are real numbers and a > b. Prove that if
ac ≤ bc then c ≤ 0.

So I figured a proof in my mind that went like this
We know that a > b but when multiplying the inequality by c we see that the inequality changes from a ">" to a "<" and this only happens when you multiply an inequality by a negative number so for ac < bc to be true c < 0 must also be true. Then for the <= case if c = 0 then ac = bc for any real numbers a,b. So ac <= bc if and only if c <= 0.

I feel pretty satisfied with my proof so I read the solution given for the example

>Assume the contrapositive, c > 0. then if you multiply a > b by c you get ac > bc so the contraty can only be true if and only if c <= 0.

It looks good but could any anon tell me if my proof is also correct? Is it sort-of correct but lacks rigor? Am I retarded? Should I drop out of pure maths already and study sociology?

>>7683708
There is no fast way, and if you do it fast you'll leave gaps along the way.

The four years of mathematical training are pretty much devoted to teach students how to write proofs. It's hard for everyone, there is no easy road
Keep solving problems. That's all there is to it.

>>7683804
>this only happens when you multiply an inequality by a negative number
You can't use that because it is implied by the proposition you're asked to prove.

>>7683804
>we see that the inequality changes from a ">" to a "<" and this only happens when you multiply an inequality by a negative number

You don't know that. In fact, that's exactly what you want to prove.

>>7683804
The real axiom is: a>b and x>0 then a*x>b*x
Take the contrapositive of the statement: a*x≤b*x then a≤b or x≤0
Since we assume a>b then x≤0 must be satisfied if a*x≤b*x.
QED

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>>7683804
>>7683849
>>7683853
>>7683879
wow wow guys, calm down. Let's get back on track

>>7683844
>The four years of mathematical training
By the gour years you mean university training or pre-university time?
>Keep solving problems.
So should I just continue studying Calculus or should I postpone it and dedicate some time to e.g. the book I posted?

>>7683896
>By the gour years you mean university training or pre-university time?
University. Pre-university math is engineering-math, boring and less focused on thinking.}

>So should I just continue studying Calculus or should I postpone it and dedicate some time to e.g. the book I posted?
If you have the time to read it, go ahead.
I personally don't like calculus because I don't like dealing with formulas and numbers, so my very personal suggestion for you is getting into some introductory topology or abstract algebra textbook, then you'll have a better idea of what modern math is about, and you'll be doing proofs that are more fun than that pesky boring calculus.

>>7683844
>>7683849
>>7683853

You are right on that, my bad. I think I'll keep reading the book and hopefully it will help me polish my high school math again so I don't make that mistake in the future.

>>7683804
Let [math]a > b \;\; \Rightarrow \;\; a - b > 0.[/math] Then, (because we assume the "if" part is true) we have

[math]ac \leq bc[/math]
[math]ac - bc \leq 0[/math]
[math]c (a - b) \leq 0[/math]
Because [math]a - b > 0[/math], the inequality sign direction does not change if we divide both sides by [math]a - b[/math] and also the operation won't be undefined.
Thus [math]c \leq 0[/math].

>>7683708
Read it. Would recommend. You CANNOT make it quick tho. The goal is to learn some core mental habits that refine the logical reasoning that you've known intuitively your whole life down to a fine point - a process which is unnatural as fuck and will take you as long as it takes you.

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>>7683708
Best book is Logic: Techniques of Formal Reasoning. It'll take you like six months to work through if you're new to logic but you go from literally nothing to proving shit like riemann integration. It's pretty crazy.

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>>7684383
what kind of philosophers study this kind of stuff?

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>>7684621
Logicians, philosophers of logic, philosophers of math, mathematical philosophers, philosophers of language, etc.

Lots of analytic philosophers often rely on logic for their theories.

pic: saul kripke

What about "Mathematical Proofs: A Transition to Advanced Mathematics" by Chartrand/Polimeni/Zhang and "Book of Proof" by Hammack?

>>7684383
you got a link for the book?

euclid elements

>>7685329
Actually kinda this, if you pair it with a book like in OP's pic to flesh out your set theoretic/logical thought about proofs. Applying the precision of the latter through the geometric intuition of the former would probably be a good way to drive home the concept of proof based math (and prolly just logical thought applied within formal systems in general).

>>7685291
First book by Zhang is a very good into proofs book. Read it going into uni and finished it first semester. Goes over most subjects, mostly baby algebra and baby analysis but still a good intro to proving things. Recommended from an advanced math soph

Honestly who gives a shit about proofs. Everything you will prove or have proven has been proven a long time ago already.

>>7685351
your mom been proven a whore long time ago already

>>7685291
Hammack is rather basic. It's something I would recommend to an enthusiastic secondary school student or a freshman not in a non-pure math field.

>>7684383
That's gay, the book torrent site I have does not have a copy.