Which math proof method is the most effective? Writing out givens,

>>7869216
There is no general purpose proof method

>>7869218
Yes there is. Sit there banging your head against the problem for hours, days, or weeks until you come up with a proof.

>>7869218
>>7869229
Said high school students who think they've ever done proofs

>>7869216
In the first example, why didn't he mention commutativity?

>>7869295
baka desu senpai

>>7869216
With any proof technique you need to write what you have, what you want, and the reasons for the steps, genius.

Read "How to prove it, a Structured Approach"

>>7869218
>What is a sound and complete Hilbert-style proof system

>>7869216
Reducto ad Absurdium

>>7869868
This.
This is my favourite one. It's easy to do.

Proof by contradiction is my go to for just about anything, because if you don't prove it, you generally stumble upon a counter-example.

>>7869917
Counterexample is much stronger than contradiction. You need to construct. The point of contradiction is that you can't construct so you need something more.

>>7869922
There isn't a "proof by counterexample."

>>7869925
>arguing semantics
Okay then. "Yes there is".

>>7869927
what exactly does proof by counterexample look like then?

This isn't semantics. There's just no such proof.

>>7869930
Exactly what is says, wiseass. Prove P by exhibiting a case where ~P is false.

The point is, if you "stumble upon a counterexample" while doing contradiction then you don't know what you're doing and you're just blindly trying shit out. You're supposed to use contradiction once you fail to construct.

>>7869933
Give an example of a "proof by counter-example"

>>7869950
Are you fucking serious?
"Prove that n^2 =/= 1 for all natural n".
Well okay, n = 2 implies that n^2 = 4 and we've proved what we wanted with a counter example.

>>7869216
>Which math proof method is the most effective?

There are basically two methods: top-down (start from the axioms) and bottom-up (working towards the axioms). The latter method is supposedly the more "natural" one and so it goes by the name "natural deduction".

>>7869930
You know about quantifieres in logic? Pretty easy shit.

>>7869930
Are you even serious or is this bait?

>>7870200
That doesn't prove that [math] n^2 \neq 1 [/math] for all [math] n \in \mathbb{N} [/math]. All you've shown is that [math] 2^2 \neq 1 [/math].

I repeat my question. Give me an example of a "proof by counter-example".

>hurr fill in da box now ur dun!!! XDDD

what a terrible way to learn how to write a proof.

>>7870823
XDDD YOU ARE SOOOOOOOO MAD XD!!!!

>>7870615
Sorry, that's supposed to say n^2 = 1 for all natural n is NOT true obviously, and everything else that I've said holds.

>>7870615
nigga,
"for all n in the naturals, n^2 = / = 1"
means that shit is true for each and every natural number. The negation of this statement is:
"There exists an n in the naturals such that n^2 = 1"
Gee, I wonder--
oh shit, 1^2 = 1
1 is a natural right???
damn son, the negation of our statement is true
oh fuck, we live in this fucked up logic system where things are exclusively true or false
oh shit m9 that means our original statement is false as shit homie o dam

this is proof by counterexample boii

>>7870615
>>7870898
oh I see what you are saying holy shit the autism

here's my statement:
"(for all n in the naturals, n^2 = / = 1) is false"

the rest follows in my other post

>>7870898
The negation of your statement is true, so your statement is false. Try proving a true statement with a counterexample

>>7870936
did you read the very next post?

>>7870944
Yes, I read the very next post. The point is that all a counterexample can show is that the original statement is false. If you want to prove a statement is true, and it is actually a true statement, then there isn't going to be any counterexample

>>7870990
If you want to prove P by counterexample, you exhibit an instance that disproves ~P. This isn't hard to understand.

>>7871016
You're describing proof by example or just proving a single case. He's being pedantic, but he's got a point.

>>7870990
>>7871016
(of course, it might be more natural to say "prove by example", but the context matters: you usually assume ~P and then disprove it)

>>7870990
Bro, the statement is that something is false
That's a statement
I proved it was true
I did it with a counter example
Don't know what else to say

>>7871098
You proved that there exists a natural number satisfying some property by showing an example of a natural number with that property. That's a proof by example.

>>7871108
No, he proved that "All natural numbers satisfy P" is false by showing a number that doesn't satisfy P. It's counterexample.

See how you're still arguing semantics?

>>7869216
By contradiction is usually the easiest.

Because it makes the most assumptions

The person saying that you can't prove a statement with a counter example is such a retard lol

>>7871108
I used a counter example to show the negation was false, therefore the statement is true

Proved it
Using a counterexample

>>7869917
Proof by counterexample is universally considered as the most shit tier proof method. It is so bad that an entire formalization of mathematics has been formed explicitly to avoid that form of argument.

The most elegant proof method is combinatorial proof.

https://en.wikipedia.org/wiki/Combinatorial_proof

The idea is that you want to prove an equation, say f(n)=g(n) where n is an integer. Then what you do is construct some mathematical entity and count it in two different ways. In the first way you get f(n) and in the second way you get g(n). Since you're counting the same thing then f(n)=g(n).

Combinatorial proofs are also considered the hardest types of proofs to come up with.

>>7871178
You can only prove very specific things this way, genius. Also how the fuck can you consider a proof by construction shit tier and a proof by construction of functions the "most elegant method"?

Confirmed for wannabe faggot

>>7871200
Proof by contradiction is not proof by construction.

It is actually negation introduction and it comes in two flavors.

>Suppose "P" is true, reach a contradiction, therefore "not P" is true
>Suppose "not P" is true, reach a contradiction, therefore "not (not P)" is false
Note that in order for the second version to work you have to use double negation elimination.
>"not (not P)" true entails "P" true.

This second form of contradiction is bad because it lets you "prove" things without ever actually giving an example. Instead you're saying
>If a thing doesn't exist then my system falls apart. Therefore the thing must exist even if no example can be found.

Getting rid of double negation yields intuitionistic mathematics. This is exactly the type of logic one deals with in type theories (such as HoTT), theorem provers, and categorical logic because in these contexts the semantics have been remixed so that saying "P is true" means that you have demonstrated a proof of P. If you can't demonstrate a proof of "not P" then it doesn't make sense to conclude that P is true. Afterall, P may be unprovable.

>>7871120
You can't prove a false statement but because its false you can give a counterexample to the statement.

>>7871301
>>Suppose "not P" is true, reach a contradiction, therefore "not (not P)" is true
fixed

>>7869216
I appreciate the thought behind your worksheet so students learn proving sooner, but the excercises are kind of dumb.

>>7869216
Coffee, and on occasion, meth.

>>7871359
Its fucking grade school material for kids to learn basic proofs in algebra. You're not going to teach them the Nippon proof of the universe.

>>7869216
Structural induction in 90% of the cases.
Natural induction on some derived property in the other 10%.
t. CS student

>>7869216
Can we talk about how this picture makes no sense.
>If AB then AB=AB
Literally what the fuck. = Is a predicate that acts on terms, so AB is not a predicate. Therefore AB cannot be true/false and the sentence makes no sense.

>>7871546
>CS
>Not using intuitionistic type theory and theorem provers.

Ha!

>>7873133
That's common core for you, apparently it's called "reflexive property". Sadly, many people actually believe what they're taught in high school about science and math only to find out that once they dive in further than some meme IB SL class, they were basically taught lies so that their dumb minds can comprehend basic theory.

>>7873146
The identity predicate does have the reflexive property on terms and it is the correct way to teach the identity predicate. Unfortunately because we don't teach it that way then retards like you don't know the difference between identity and equality.

However that is not the issue present in the picture. The issue is that AB isn't something you can assume.

It's like saying:
>Theorem: Prove that if 5 then 5=5.
>Proof:
>Suppose 5 is true.
>....
>Then 5=5 is true.
>Thus if 5 then 5=5.

The problem is that 5 isn't something that can be true or false. It's not a predicate. The above makes no sense.

If on the other hand the statement said
>If AB exists then AB=AB
That's something else entirely and would be provable through the reflexive property of the identity predicate.

Only retards who can't into abstract math think common core is shit. Fuck you, you retard.

>>7873158
"Prove 5 = 5"
>what are abstract concepts

>>7873163
That's not the same statement. The statement is "IF 5, then 5=5". You obviously do not understand abstract concepts since you have made such an obvious mistake.

File: 1455696876318.png (465KB, 600x531px) Image search: [Google]

465KB, 600x531px

>>7873173

contradiction is one of my favorite tools in a proof.

Almost all my proofs are proof by contradiction

>>7873232
He's right. You make the Donald look bad with your faggotry

>>7871136
what he is saying is that you cant prove a statement as true by showing a counterexample where the statement is false
i.e. you cant prove (all x)(P) by showing (exists x)(not p)

which is true

or atleast i hope thats what he's saying i hope hes not retarded

>>7873232
Retarded weeaboo manchild get out of this board

>>7874926
but you can disprove (all x)(P) with (exist x)(not P)

>>7869216
>Which math proof method is the most effective?
Answering that would require... The set of all math proofs.

Why haven't I formally performed this analysis yet?

>>7869216
Sounds idiotic.

>>7869218
this

>>7869295
kek

>>7869216
Just use the reflexivity of equality. A longer, but more rigorous way would be to use the reflexivity of equality, the closure of multiplication, and the well-defindness of multiplication.

>>7874865
That's bad form. You can almost always do a proof by contrapositive instead, which is better.

>>7869216
>I'm smarter than all my peers

I bet you never talk so you can'take ever hear how stupid and average you and your ideas really sound. Autism will be the death of radical science