/sci/ - Science & Math

>>11557097
I think that you have to use the properties (A>b but a has to be positive) but im a Biofag so this is just a bump

>>11557139
What do you mean?

>>11557097
Send the question

>>11557097
> I understand the math
> but I can't prove anything
pick one

>>11557294
I don't think that's the problem. Maths has it's own unique names for different kinds of proofs, even if they're normally used by people in general. This special naming makes it confusing-- even if you can prove something, you don't know if you're using the correct/expected mathematical paradigm for proving shit.
>>11557097
Check this out: http://logic.stanford.edu/intrologic/public/index.php

>>11557290
It's pretty much every question. From Algebra I to II I've only been doing exercises like "solve ..." or "find ..." or whatever. I haven't dealt with any proofs so I wonder if there is a bridge I'm missing that I need to move on to Calculus from Algebra because I have no idea how to prove things
>>11557294
It's all basic stuff like inequalities like |a + b| <= |a| + |b|, I could solve this inequality easily with inputs but I can't prove this case i.e.
>>11557312
How is propositional logic going to help me with proofs? No one told me you need to know Logic before moving on from Algebra to Calculus

>>11557097

Logic is form, Calculus is content. You need to be able to distinguish between the two. In logic you are evaluating whether or not an argument is valid. This is equivalent to showing that there is no interpretation of the argument in which the conclusion is false while the premises are true. This is achieved by forming a conjunction between all of the premises, and then further condensing the argument into an implication from the newly conjoined premises to their conclusion. If the implication doesn't hold, i.e. there is an instance in which the premises are true but the conclusion is false, then the argument is invalid. If the argument is valid, it can either be vacuous or not. Ideally you want to make non-vacuous valid arguments.

Since the logic we use for writing proofs is bivalent, statements can either be true or false. This is critical when writing your proofs because the definition of the terms you are studying need to be understood in a way such that you can decompose compounds statements into their constituent simple statements that are either true or false, also known as atomic statements.

Statements like "x <= 3" is not an atomic statement because it uses a disjunction connective ('v' also called "wedge") to form the composite statement "(x < 3) v (x = 3)," which is composed of atomic statements, "x < 3," and "x = 3."

>>11557326
>>11557334

To summarize, proof writing can be boiled down to showing that an implication between the premises of an argument and the conclusion is a tautology. Often the fastest way to prove this is by only looking at the cases in which the conclusion is false- since those are the only cases in which the argument could be invalid. Within the interpretations in which the conclusion is false- if there are no interpretations (truth value assignments to the variables of your statement) in which the premises are true, then the argument is valid.

So essentially, you are really just looking at interpretations in which the conclusion of an argument is false, and then trying to find a true premise. Since the premises are connected by a conjunction, this is actually quite easy because conjunctions are true only if all its constituent statements are true. So you're really only interested in interpretations in which all the premises are true and then conversely looking for a false conclusion.

So the key points are:

1. Convert all the definition of terms from calculus into atomic statements
2. Convert all theorems into implications between premises and conclusions
3. Identify interpretations (truth-value assignments of the atomic statements) in which all of the premises are false
4. Find a false conclusion among those interpretation

If you can't- the argument is valid. If you can, it's invalid.

To summarize, proof writing can be boiled down to showing that an implication between the premises of an argument and the conclusion is a tautology. Often the fastest way to prove this is by only looking at the cases in which the conclusion is false- since those are the only cases in which the argument could be invalid. Within the interpretations in which the conclusion is false- if there are no interpretations (truth value assignments to the variables of your statement) in which the premises are true, then the argument is valid.

So essentially, you are really just looking at interpretations in which the conclusion of an argument is false, and then trying to find a true premise. Since the premises are connected by a conjunction, this is actually quite easy because conjunctions are true only if all its constituent statements are true. So you're really only interested in interpretations in which all the premises are true and then conversely looking for a false conclusion.

So the key points are:

1. Convert all the definition of terms from calculus into atomic statements
2. Convert all theorems into implications between premises and conclusions
3. Identify interpretations (truth-value assignments of the atomic statements) in which all of the premises are true
4. Find a false conclusion among those interpretation

If you can't- the argument is valid. If you can, it's invalid.

>>11557359

So to use the statement from your post, we would do something like the following.

For all real numbers, a, b.

|a + b| <= |a| + |b|

This statement would translate into the following logical statement:

(|a + b| < |a| + |b|) v (|a + b| = |a| + |b|)

This is a disjunction. You simply need to show that the disjunction can't be false. Disjunctions are only false when both constituent statements are false. If you can show that one of the constituent statements is always true, then it cannot be the case that both constituent statements can both be false. Let's make it easy and take the obvious one.

|a + b| = |a| + |b|

If you show that this can never be false, then the disjunction can never be false, and therefore the statement is a tautology and the theorem is true.

>>11557326

Another thing that makes proofs unique from "regular math" is that in proof writing you don't instantiate variables with actual numbers, but instead the generalization of whatever number you would ordinarily instantiate it with. For example, in proving that |a + b| <= |a| + |b|, you wouldn't compare a = -2 to b = 3, you would compare a < 0 and b > 0. You are interested in general results, not specific results.

>>11557359
>>11557334
I don't understand any of this and how does it answer my question? I doubt anyone was told about all this when going from Algebra to Calculus
>>11557497
That much I can understand, of course, but it does not answer my question in the slightest

>>11557513
https://math.stackexchange.com/a/307384

The computations that you do in high school isn't real math and isn't real algebra. Algebra is about structures. Pic related is real algebra.

Work your way through this series

https://m.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6KzmkUQBQ8TSWVX

>>11557513

You are fucking cancer.

>>11557581

This kid is retarded and just wants to copy and paste proofs from stackexchange, not understand how you write them and why they work that way.

>>11557097
I've been struggling with finding an unified subject that covers the validity of statements and how to write them to form a proof, the closest to that seems to be Aristotle's logic. https://www.youtube.com/watch?v=hj7u3G_K6TA

>>11557326
>how is propositional logic going to help me with proofs

Ban him.

>>11557600
>>11557604
>>11557581
>>11557608
I'm literally a highschool student who just finished doing algebra with only computations and formulas, now I am introduced with proofs in Spivak's book and have no idea how to do these exercises cause I lack the knowledge of how to prove things. And you guys solution is to go through propositional logic?
I have almost never heard of people who do Logic (let alone a course in it) after Algebra when going into Calculus, but you guys talk like that is a common-sense prerequisite, I don't get it.

>>11557604
So here is a bit of what I've gathered, for every math proof you need to mention the "rules" (premises) you will be using to lead to a conclusion, then you compare that conclusion to the result (which i imagine is called conclusion too) in the exercise. e.g.

Prove that 2 < 0.
Premise: In a decimal system, we have that 0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8 < 9 ...
From this, we observe that 0 < 1 and 1 < 2, then we can conclude that 0 < 2.

There, proved. There is also a name for the form "if A is less than B, and B is less than C, then A is less than C" that I forget at the moment.

>>11557581
sonic the hedgehog 2020 official trailer

You need to get used to it. Spivak didn’t write his books for self teaching. The way those books work is you hit a brick wall, a mentor (professor, his assistant, someone helping you) shows you how to do it, you apply that method to the next few exercises, you hit another brick wall, repeat. It’s why his problems aren’t really gradual in difficulty. Instead you have jumps which require learning a new technique and then the difficulty remains the same so you can practice that. Then another jump and so on.
I suppose the role of the mentor could be somewhat sidestepped if you find the solutions to the exercises with some kind of explanation.

>>11557651

You don't need propositional logic to learn calculus, you need propositional logic to learn proofs, which is what you were fucking asking about. if you are writing proofs, you need propositional logic. end of story. if you don't want to do calculus with proofs, read stewart. there's nothing wrong with doing it without proofs but you asked about fucking proofs and then when we gave you the fucking answer you said "that doesn't make sense." Too bad. If you can't figure out why you need logic to write proofs, no one on this board can help you.

>>11557683

lmfao just saw that

>>11557724
That's insane, my friend goes to uni and before they took analysis class they had a small course in proofs showing how to prove with induction and things, they never used propositional logic and went on fine without it

I will read Vellman's "How to Prove it" then before I attempt calculus

A big problem with proofs for self taught students is that it’s extremely difficult to self grade the exercises. Problems involving calculations are friendlier to those students because they can check the solutions very easily. Proofs not so much.

>>11557651
you first mistake was to ask for advice on this god forsaken board. Take a look at the stackexchange link i posted to see an example how an elementary proof lookslike. Try to build your intuition from doing exercises like this.

>>11557747
If you want to "learn" calculus without proofs use "Calculus: Early Transcendentals" by Stewart

>>11557651
Might be better for you to study a computational book rather than a proof-based book like Spivak. Most people usually go through their calculus course and learn how to perform the computations first and then learn how to do proofs as their mathematical maturity develops.
You could use Stewart Calculus or Calculus and Analytic Geometry by Thomas & Finney.

>>11557748
>checking the output of a function is easier than checking the main connective column on a truth table

No. You are doing the same amount of work in both situations.

>>11557747
>small course in proofs
>never used propositional logic

Hate to break it to you but your friend is retarded anon.

>>11557755
I do want to learn Calculus with proofs, I hope to one day enroll into pure math university
I just didn't know that to learn calculus with proofs, you need to know proofs as a requisite. Many people here recommended Spivak but none of them actually said I needed to know proof beforehand.
>>11557748
Makes sense
>>11557750
Hmm learning by example, I will try that, but the thing is Spivak also gives examples of proofs and I can follow them cause he has good definitions, but then when I'm asked to do them im like OMG
>>11557774
That I find sad to hear, so I have to do an entire computational calculus course, only to repeat it but with proofs afterwards? Doesn't that seem like a waste of time unlike just doing the proof-based calculus in the beginning?

Should I try to do Linear Algebra instead to increase a bit in maturity then try Spivak later? Or is it even possible to do Algebra to Linear Algebra?

>>11557814
I just checked most universities have the math curriculum built like this:
Calculus
Introduction to proofs
Real Analysis I
Algebra I
Algebra II
Topology I
Topology II
How can you say you need propositional logic when most universities don't even have the subject but instead a course in proofs? (where you learn induction, contradiction, etc)

>>11557513
> I doubt anyone was told about all this when going from Algebra to Calculus
Because this is an advanced calculus / intro to analysis textbook not typical calculus. If you're dead set on reading it (or want to become a math major) you might want to start by reading book of proof, how to prove it or a transition to advanced mathematics.

>>11557581
thanks for the link bro!

Apostol's Calculus (from the first chapter) as an introduction to proofs

>>11557892
Euclidean geometry also works. Have you ever seen a proof of the pythagorean theorem?

>>11557097
I went through the exact same thing. I got tired of being made fun of for asking stuff here and eventually accepted I'll have to do a lesser calculus book first, although I'm doing none now because I have more immediate needs. I know ideally I should've read the book of proof and then continued with Spivak, but I still have no patience for this shit given my area likely won't require me to use it frequently and I'll end up forgetting it unless I'm always doing exercises. Don't get me wrong, I still want to learn it, but priorities.

>>11557326
Just go through the cases. If a, b are positive, what is the inequality. If a,b are negative, what is the inequality. If the signs are different, what is the inequality. Exhaust your cases.

>>11557097
>I understand the math but I can't prove anything yet
Prove it.

>>11557748
Maybe I finally have the vaulted "mathematical maturity" now but I usually know from the moment I finish, if my proof is satisfying or not

Anyway Spivak has a solution manual

>>11557814
>he thinks people write MATHEMATICAL proofs using truth tables
Have you taken a proof-based math course beyond intro to discrete math? Nobody does this. Literally nobody.

>>11557747
>I will read Vellman's "How to Prove it" then before I attempt calculus
Yes, just do this. Don't bother with calc until you actually get there in school

>>11557326
Get Discrete Mathematics from Rosen, if you do everything from the first few chapters you'll have a really solid base on proof making (and it's made for dummies too).

>>11557097
Just read How To Prove It.

Landau & Lifshitz

>>11557097
>I understand the math but I can’t prove anything yet
JUST LOL AT YOU

>>11557934

you are a fucking troll

WTF are you talking about anon? Did you already go over trigonometric identities in your algebra curriculum? I cannot comprehend how you could possibly go through with doing proofs with calculus if all you have is an algebra background. Generally, trigonometry and pre-cal are the first classes to take before Cal I here in the states. Linear algebra is slowly being taken out of public institutions. So if you have no comprehension of the relationship between using geometry to form results then you couldn't possibly know where to start with some of those exercises.