(G, *) is a group IFF

(G, *) is a group IFF
>* is an associative binary multiplicative operation
>for each a, b of G: a*x=b; y*a=b has a single solution of GxG

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>posting high level mathematics on a technology board

It's a CS board and CS implies proof-based mathematical education. I want to see how many people are not consumerist LARPers. I'll post this on /sci/ soon.

Group theory is not CS you nerd. Most CS majors take baby calc, baby stats, and discrete math.

> high level math
the absolute state

How does one learn first order logic and catecory theory without a basic yet concrete foundation of morphisms and operational subsets?

This is a technology board, not CS only.

By taking a logic class. Most of the time it's offered from the fucking arts department. We can teach literal babies induction and you think it's hard for someone of average IQ to understand FOL?

I don't expect arts classes to describe and apply the method of resolution or to prove satisfiability. Because the former is computationally impossible and the latter requires a level of mathematical maturity.

False for any group G with a nullity element.
For example
G = Z, a= 0, b= 0 has infinite solutions

0 has no multiplicative inverse element, thank you for your attempt.

Define * as multiplication with 0 * 0 =1. Go home

For each a:
1 * a = a
(0 * 0) * a = a
0 * (0 * a) = a //assoc.
0 * 0 = a
1 = a
Are you fucking crazy?

G = { 0,1}

Or even easier if * = +

How is that G with your * a counterexample?
And this?

I'm sorry do you not know what iff means?

>implying operator symbols have any meaning by themselves

bleesed post, a good change from all of those win vs linux/intel vs amd posts

good job OP

Sit down and write down the proof

Tell me what iff means and if you still don't see why you are wrong I'll explain it harder to you.

this is an ECE board you fuck

See
Stop shitposting, I already proved your counterexamples wrong and you refuse to comment further.

based

Aww it's very easy, I'll even get you started so you can not embarrass yourself in front of class tomorrow.
"Both the implication and it's ________ must be true."

That's not even true, F F is T. And you didnt write a proof. F-.

Whats wrong with this fucking board

You don't need those to learn FOL. Maybe if you want to study philosophy or math postgrad, but I wouldn't expect that of a CS undergrad. I have no clue what kind of backwards CS program you studied at, but this isn't a prerequisite for anything you'll see in a 4 year program.

Lawyers learn "logic" too. Take a look at some LSAT logic games and you'll see just how basic everyone outside of stem is when it comes to math and logic.

Wow I guess we have to back up even further for you to understand how utterly wrong you are. Did you poorly copy this problem from somewhere and pose it to us thinking you would look smart?

What do you think an implication, in math.

Well maybe some people want a masters or maybe a doctorate? There are four computer majors here, from low difficulty up: information systems, software engineering, computer science, and informatics (CS + applied maths majors). Im an average at best student of the latter and I have the impression Jow Forums is full of great programmers.
you're having a schizo moment because you cannot factually refute me. In fact, this is quite literally the first problem in my textbook on groups.

So I was correct when I said you poorly copied it from somewhere...
How interesting.

Yes very interesting, now demonstrate me your proof.

We've already been here. For you to understand why you are wrong, tell me what you think an iff is.

No amount of proofs are going to help you if you are lacking a basic understanding of terms.

this isn't a problem, it's a definition

You didn't refute me in any way. You represent Jow Forums at the moment so stop acting like a retarded shitposter. It's a 5-line proof in each direction.

Jokes on you if actually thought Jow Forums was full of great programmers. Doubly so if you thought asking them about groups would call them out. Honestly whatever the hell you're learning is going to pretty useless if your goal is to be a "good" programmer. I legit dropped logic II cause that shit bored me to death and I'd rather take another option more relevant to my degree. I also wanted to get a minor in math but went back on that one pretty quick. I'm employed now, so I think I turned out alright

Also masters/PhD candidates are pretty rare anywhere but a university campus. I don't know why'd you expect more here.

Please leave

Except the for the counterexamples yeah you've been right the whole time.
And refusing to answer basic definition questions but don't worry about that.

You should be able to solve this simple problem first OP:

If a group has r generators and exponent n , is it necessarily finite?

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only if (r, n) = 1

>multiplicative operation
This term has no meaning. Try to read your course before asking us to do your homework.

Not technology.

So you're saying n*a = a^n?

Nobody cares about category theory except the functional programming autists, if you got it taught in undergrad then you got meme'd

FOL with resolution etc. is taught in the first semester of CS undergrad here

>babby's first abstract algebra problem

>IFF
The fuck is this t. cs

If G is the empty set then its vacuously true.
If G is the set with only zero and 1 then it also holds

oups I meant set with only 0. No 1. Just the 0.

>it's true because I say so
Show the other direction.
>it also holds because I say so
Not convinced.

I forget all my math when I don't practice it/do it in uni.
Its summer break now so...
Also Im not doing your homework for free

Shant be solving your applied linear algebra homework for you

I ran out of the space. So the neutral element is unique from my choice and the inverse is unique (assume the opposite and done). And the associativity.... It must be a group.

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Would khan academy help me to learn more math like this?

Yeah I'm not reading that ugly, schizophenic mess.
No, long and dry explanations made for high schoolers. Get a good book that starts with "Introduction to" and doesn't end with "for dummies".

Look at him and laugh.

>multiplicative
That word doesn't really have a meaning in the context of groups. For many groups the operation is interpreted as multiplication but there's plenty of additive ones (and yet others).
>has a unique solution (x, y) in GxG
ftfy

Anyway: depending on your definition of operation, the closure property may or may not be missing from your definition. Usually that property is not included.

Then it still seems a bit off. Deriving left and right neutral elements is easy but showing they're the same seems not. I guess I'll look for a pen and some paper.

How is it possible left and right neutral elements to be different?

In groups it's not. That's the point. For non-group examples with a property like that I'm sure one can build something with matrices.
But for the given definition no assertion about their identity is made.
Same problem then with left and right inverse elements.

In wikipedia and my text book it's implied left and right are the same. Where can I read more?

It's usually defined that way. A weaker definition of a group requires only right (or left) neutral and inverse elements for which can then be shown that they have the same property on the other side as well. Left for one and right for the other doesn't work.
I heard about this in a lecture and also have an algebra book with the same theorem. Don't know where you'd find it online.

Thanks, I'll keep it in mind. Care to write OP's proof?