Understanding the Logic of Quantifiers: A Guide for Mathematical Proofs

[Obliv]

I'm new to proofs and I'm not sure from which assumptions one has to start with in a proof. I'm trying to prove the generalized associative law for groups and if I start with the axioms of a group as the assumptions then I already have the proof.

From what basic assumptions should one start with in proving something? I'm thinking of starting with the definition of an equivalence relation. Would that even help with proving something like (a R b) R c = a R (b R c) for all a,b,c $\in \mathbb{G}$?

 

[fresh_42]

I'm new to proofs and I'm not sure from which assumptions one has to start with in a proof. I'm trying to prove the generalized associative law for groups and if I start with the axioms of a group as the assumptions then I already have the proof.

From what basic assumptions should one start with in proving something? I'm thinking of starting with the definition of an equivalence relation. Would that even help with proving something like (a R b) R c = a R (b R c) for all a,b,c $\in \mathbb{G}$?

What is "the generalized associative law" and how is "group" defined? There are basically two different approaches: as an enumeration of properties of the group operation (neutral element, inverse element and so on) or as the solutions of certain equations like $xa=b$.

This means: it is always helpful to state exactly what is known (or given) and what has to be shown. The more precise these statements are the easier it is to find a proof. Of course not always, but often. In any case it helps reading a proof and for sure the debate on a platform like PF.

 

[Obliv]

Well a group is defined as an ordered pair of a binary relation R and the set $\mathbb{G}$ as (R,$\mathbb{G}$) and the generalized associative law is that $a_1 ~R~ a_2 ~R ~a_3~ R ...R~ a_n \in \mathbb{G}$ this is independent of how you place brackets around the elements.

If I can prove this for a few elements in the group then I can do this with all of them through induction right? Without using the definition of a group (having to be associative, there must exist an identity element, and there must exist an inverse for each element) how can I do this?

 

[Obliv]

Actually, It is quite easy to do this if we assume the relation is equivalent (since symmetry means commutativity and proving associativeness follows very quickly) but under the definition of a group it doesn't have to be equivalent so I have to find another way to prove the associative property darn

 

[fresh_42]

Well, you aren't allowed to use commutativity since non Abelian groups also obey the associative law. But induction sounds good.
In this case it is really more an exercise in precise wording than it is to find the proof.

 

[chiro]

For proof's I'd recommend looking at it from an intuitive view before trying to formalize things.

Langauge is often suited to ones intuition and mathematics is no exception to that rule.

If you can get the intuition (which may draw from all sorts of visual examples and physical intuition) then the formality will get easier.

The formalities are necessary to make everything precise and consistent - but the intuition of ideas, concepts and information should be a primary point - not a secondary one in my opinion.

Also - what mathematicians (and those who use mathematics) do is to learn how to combine information to get to the resolution they are intending to make (not so much the answer they are expecting - but the resolution in regard to that answer).

Learning that takes a bit of practice - but it can be done and has been done by many who have studied and worked at the subject of mathematics and its applications.

 

[Stephen Tashi]

If I can prove this for a few elements in the group then I can do this with all of them through induction right?

The technique wouldn't be called "induction".

Since you asked for general advice, I'll offer this very general advice.

Underlying mathematical proofs is the use of logic. It is the type of logic that deals with quantifiers, such as "for each" and "there exists". Also underlying mathematical proofs is "legalism", i.e. a hair splitting approach to what mathematical definitions say.

Some people develop the skill of doing mathematics (and other legalistic activities) without any formal training in logic. If you find yourself bewildered by mathematical proofs, I suggest you at least glance at textbook that deals with the logic of quantifiers in its later chapters. Look at a book that uses many non mathematical word problems - problems that begin with statements like "Given that each student has at least one friend and that all students who have friends have a best friend, prove that ...". You don't have to study an entire book in detail, you only need to get the general idea of how to deal with the logic of quantifiers.