Axioms Definition and 181 Threads

Maybe this is a dumb question. I'm a bit tired right now. :smile: What is a "theory" in mathematics, and what kind of statements can we call "axioms"? To be more specific, is "group theory" a mathematical theory, and if yes, what are its axioms? Should I think of the definition of "group"...

Well, I'm supposed to prove 0v=0 It is stated that I'm only allowed to use the following axioms. let a,b,c be vectors and V is a vector space, then 1)a&b is in V then a+b is in V 2)a+b=b+a 3)a+(b+c)=(a+b)+c 4)0+a=a+0=a 5)a+(-a)=(-a)+a=0 6)a is in V implies ka is in V...

Hey guys, I need to prove a few theorems about vector spaces using the axioms. a) Prove: if -v = v, then v = 0 b) Prove: (-r)v = -(rv) c) Prove: r(-v) = -(rv) d) Prove: v - (-w) = v + w where r is a scalar and v, w are vectors.

I assume someone has figured this out... If so, would anyone mind explaining it to me?

Homework Statement Prove that a set G, together with a binary operation * on G, satisfying the following three axioms: A1) The binary operation * on G is associative A2) There exists a left identity element e in G such that e*x=x for all x in G A3) For each a in G, there exists a left...

:rolleyes: Do a mathematician believe in AXIOM like some people believe in GOD People who believe in GOD need not to proof GOD, is that like the way mathematician do with AXIOM ?

Axioms, as long as they remain axioms: 1) Are not falsifiable by experiment 2) Are not verifiable by experiment 3) When required by theories, are subject to questioning by non-scientists and conspiracy theorists 4) Make theoretical scientific notions which need them speculative rather than...

Homework Statement One of the fundamental axioms that must hold true for a set of elements to be considered a vector space is as follows: 1*x = x I was given a particular space: The set of all polynomials of degree greater than or equal to three, and zero, and asked to evaluate whether or...

Wiki says that a sigma algebra (or sigma field) is a subset \Sigma of the powerset of some set X satisfying the following axioms 1) E\in \Sigma \Rightarrow E^c \in \Sigma 2) E_i \in \Sigma \ \ \forall i \in I \Rightarrow \bigcup_{i\in I}E_i \in \Sigma (where the index set I is countable) Am...

Hello, Can you please help me with the questions listed below. I would like to get hints on how I can solve them. I have listed first the axioms and then the questions at the bottom. Axioms: --------------------------------------------------------- A plane consists of: -two sets P...

It is known that "the integers under addition" form a group, that is (Z,+). I have always wondered how to actually proof that (Z,+) is a group? Definitions for a group from wikipedia: http://en.wikipedia.org/wiki/Group_(mathematics)#Basic_definitions I'm especially interested in two things...

Are basic principles like those of Newton's laws of motion considered as physics axioms on which you expand to physics theorems? Where can I find a list of physics axioms?

Another thread got me thinking... Everyday I take axioms for granted, eg. muliplication, addition, ordering of reals. From the pure point of view, what axioms are the most important (most used) ones? Wikipedia has a list: http://en.wikipedia.org/wiki/List_of_axioms However, I'd like...

Hi! I'm a high school student and I've been interested in Logic for some time... Although I read some books and acquired some knowledge, I still have one question that remains unanswered in spite of my hard work... My tutor told me that the three axioms of Propositional Logic (see them for...

Hi. I'm reading a simple introduction to groups. A group is said to be a set satisfying the following axioms (called the 'group axioms'): 1) Associativity. 2) There is a neutral element. 3) Every element has an inverse element. 4) Closure. My questions is simply: why are they...

I've read several QM texts which list five or four axioms for Qm, from which the rest is derived. I was wondering what might the axioms of Classical Physics be. I'm assuming one of them is: \delta{S} = 0 What might the others be?

So does everybody know the Axioms of Addition and Multiplication? They are too long to type, but they are listed: A1, A2, A3, A4, A5, M1, M2, M3, M4, M5 and the distributive law, DL. anyways, I want to prove: 1. (-x)y = -(xy) and 2. (-x)(-y) = (xy) using ONLY the axioms of additon and...

Second week in Linear Algebra... My homework involves of identifying all failing Vector Space Axioms for various sets of vector spaces. I did fine with a "regular" set like (x,y,z) which has an operation like k(x,y,z)=(kx,y,z). I have worked through all 10 of the axioms, comparing left sides...

X axioms... lets assume we have a consistent system which have X axioms/definitions, can we infer (or deduce) somehow from this given, how many theorems/lemmas are there in this system? (or deduce the maximal theorems/lemmas that could be proved in this this system). my initial answer would...

How do separation axioms carry over to subspaces? Some are clear -- it's easy to see that if any two points of a space X are separated by neighborhoods, then the same must be true of any subset S of X. But what about the nicer ones? Is it true that if S is a subset of a normal space, that...

In my probability class we were given two versions of probability axioms which are equivalent. Let S be the sure event, A and B any arbitrary events, I the impossible event. I will use u to denote union, and n to denote intersection: Version 1 1. P(S)=1 2. P(A)>=0 3. If AnB=I, than...

Hello. I am working on a paper detailing Godel's Incompleteness Theorm and I came across this statement. "An axiomatic base where all the axioms are true cannot prove anything to be false." Is this correct?

I'm not studying algebra yet, I just happened to notice this and am curious. Mathworld's entry for the field axioms doesn't include closure axioms, but I have seen other authors include closure axioms in the field axioms. Does anyone know why this is or what difference it makes? Can closure be...

Okey, this might be a silly question. I know that theorems are deduced logically from the axioms. But I was just wondering is it possible to deduce an axiom from the theorems? In another words work backward, assuming the required theorems are known.

How could I show that (-m)(-n)= mn? The only thing I am allowed to use to prove this are the 5 basic mathematical axioms which allow for the commutative property and associative propery of the binary operations multiplication and addition;there exists an additive inverse for each integer, 1 is...

Loop and Paden have brought up Hardy's axioms of quantum theory what do you think is usually meant by quantizing a classical (non-quantum) theory? And how does this connect to these axioms of what a quantum theory ought to be Here is a mainstream summary description of what quantizing...

what are they ? i know they are related to quantum theory.

A mathematical system S is defined as S={E,O,A}. A is the set of axioms describing the system. Is the definition of E considered an axiom? For example, if I want E={a,b}, then in the set A, do I write A={...,E={a,b},...}? Also, is the definition of O an axiom? Say O={~,V} and then I define...

I came across this site: http://mally.stanford.edu/tutorial/sentential.html It lists four axioms of "sentential" logic. I first would like to know if there are other axioms not listed here. Wouldn't you need some axiom like if P is true, then ~P is false? It seems difficult to prove the law...

what are them?

what are they?