Axioms Definition and 184 Threads

I am trying to shorten and generalize the the definition of a vector space to redefine it in such a way that only four axioms are required. The axioms must hold for all vectors u, v and w are in V and all scalars c and d. I believe the four would be: 1. u + v is in V, 2. u + 0 = u 3. u...

Self explanatory. The Cantor's theorem which am referring to is that the cardinality of the power set of any set is greater than that of the set.

My assignment is like this: 1.give an example of a space X and a subspace A of X s.t X satisifes Sep and A doesnt. 2.give an example of a continuous and onto function f:X->Y s.t X satisifies S1 but Y doesnt. 3.give an example of a continuous and onto function f:X->Y s.t X satisfies S2 and Y...

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?