Axioms Definition and 181 Threads

I am looking for a book that starts at the standard ZFC axioms and progresses to the point where some recognisable non-trivial mathematical statement is proved. By recognisable I mean something that you may encounter in school/early university level and is not purely set-theoretical (e.g...

Problem: Let $F$ be a field and let $G = F \times F$. Define multiplication and addition on $G$ by setting $(a, b)+(c, d) = (a+c, b+d)$ and $(a, b) \cdot (c, d) = (ac, bd)$. Does this define a field structure on $$G$$? I know field axioms but I'm unable to apply them to this problem. How do...

According to wikipedia AQFT needs test functions so that the fields are distributions smeared on these functions. I'd want to know what are these test functions. I read in Haag's book that they are fast decreasing functions defined on space time. They belong to the set S of Schwartz functions...

I've been asking myself a few questions lately regarding the nature of a theory. It can be any type of theory. I hope someone can answer these to a degree. The questions are: Can a theory have less than three axioms? Is three the minimum for a theory to make sense? Is the statement "The less...

Homework Statement Let V be the set of all ordered pairs of real numbers. Suppose we define addition and scalar multiplication of elements of V in an unusual way so that when u=(x1, y1), v=(x2, y2) and k∈ℝ u+v= (x1⋅x2, y1+y2) and k⋅u=(x1/k, y1/k) Show detailed calculations of one case...

From Apostol's Calculus Volume I, "Area as a Set Function" 1. Homework Statement : Right triangular regions are measurable because they are constructed from the intersection of two rectangles. Prove that all triangular regions are measurable and have an area of the product of one-half, their...

Homework Statement Let F_{2} = {0, 1} denote a field with 2 elements. Let V be a vector space over F_{2}. Show that every non-empty set W of V which is closed under addition is a subspace of V. The Attempt at a Solution subspace axioms: 0 elements, closed under scalar multiplication, closed...

I have four axioms and I am stuck trying to prove the independence of these axioms. Axiom 1: Each game is played by two distinct teams. Axiom 2: There are at least four teams. Axiom 3: Exactly six games are played. Axiom 4: Each distinct team played once against the same team. I've justified...

I was looking for explanation why x^0=1. thread https://www.physicsforums.com/threads/my-simple-proof-of-x-0-1.172073/ is locked and i did't found solution in it from axioms. People using exp(x) and log(x) and xa-a=xax-a as given. If you have xa-a=xax-a for a∈ℤ and x∈ℕ+ then there is no...

Let S={x ∈ R; -π/2 < x < π/2 } and let V be the subset of R2 given by V=S^2={(x,y); -π/2 < x < π/2}, with vector addition ( (+) ). For each (for every) u ∈ V, For each (for every) v ∈ V with u=(x1 , y1) and v=(x2,y2) u+v = (arctan (tan(x1)+tan(x2)), arctan (tan(y1)+tan(y2)) )Note: The...

In all the topology textbooks I used in school, the open set axoims specified 4 conditions on a set S: (i) S is open (ii) empty set is open (iii) arbitrary union of open sets is open (iv) finite intersection of open sets is openI noticed on proofwiki, that (ii) is omitted. I was curious if...

Homework Statement Find examples of subsets in a coordinate space where: (a) closure addition axiom doesn't hold but closure multiplication does hold, (b) closure addition axiom does hold but clouser multiplication doesn't hold, (c) where neither hold. Homework Equations None in...

First day of school and I don't have much work. So I got bored and read ahead a bit. From a textbook my prof wrote: Can someone explain why they used the "$\equiv$"? I think it means "equivalent", but I'm not sure, but when are times you want to use that symbol rather than "equals"? What's the...

College sophomore here and just wanted to ask for an explanation of terms that I have come across in my math and philosophy courses thus far, but without direct teaching of them. I've seen these terms used in forum discussions and mentioned off-handedly by profs., but don't have a specific...

Homework Statement Suppose A is a subset of B. Using the three axioms to establish P(B)≥P(A) Homework Equations Axioms... 1. For any event A, P(A)\geq0 2.P(S)=1 3.If events A1,A2... are mutually exclusive, then P(\cupAi)=\sumP(Ai) The Attempt at a Solution From the venn...

I'm wondering how one solves for the quantum harmonic oscillator using matrix methods exclusively. Given the Hamiltonian \hat{H} = \frac {1}{2m} \hat {P}^2 + \frac {1}{2} m \omega ^2 \hat {X}^2 and commutator relation [ \hat {X} , \hat {P} ] = i \hbar \hat {I} is that enough for premises...

Homework Statement Show that C[a; b], with the usual scalar multiplication and addition of functions, satises the eight axioms of a vector space. Homework Equations Eight Axioms of Vector Space: A1. x + y = y + z A2. (x+y)+z=x+(y+z) A3. There exists an element 0 such that x + 0 =...

In math, we have axioms that we assume to be true. We don't have proofs for it. Similarly english or any other natural language attempts to describe the world using words, alphabets etc. So there must be some axioms, right? Everything cannot be described / defined. But obviously, we can...

Can somebody please give me a very introductory list of the Zerkmelo-Frankel Axioms? Nothing really technical, just basically what each one means. Thanks!

In a Goemetry book i read the following two axioms. 1) There exist at least two different points on each straight line 2) There is exactly one line on two different points But the 2nd doesn't imply the 1st axiom??

Think for example of the torus as a square with the proper edges identified. Viewed like this (i.e. using the flat metric), it clearly has zero curvature everywhere. More specifically, it seems Euclid's axioms are satisfied. But however we have non-trivial topology. So what's up? Or is...

I've already asked somebody through email this question, so I'll copy and paste part of my email: Basically, I'm wondering why doesn't it fall from the other axioms, and if it does in fact not fall from the other axioms (which it apparently doesn't), why the axioms can't be slightly modified...

Please refer to the attached image. for problem 1a) i can see how this makes intuitive sense, however the hint confuses me. When we are told that $A_n \subset A_{n+1}$ why would the hint say to attempt to express $A$ as as a union of countably many disjoint sets, when it is defined not to be...

Quick question about the metric space axioms, is the requirement that the distance function be positive-semidefinite an axiom for metric spaces? It seems that it can be proved from the other axioms (symmetry, identity of indiscernibles and the triangle inequality). BiP

Given the following axioms: For all A,B,C: 1) A+B=B+A 2) A+(B+C) =(A+B)=C 3) A.B=B.A 4) A.(B.C) = (A.B).C 5) A.(B+C)= A.B+A.C 6) A+0=A 7) A.1=A 8) A+(-A)=1 9) A.(-A)=0 10) Exactly one of the following: A<B or B<A or A=B 11) A<B => A.C<B.C 12 1\neq 0 Then prove using only the...

In math, we define new concepts and have certain axioms regarding the concept that are believe to be true. Now how do we know that the given axioms are enough to give a proof to any problem? I hope this makes sense.

Axioms are: - 1) P(E) >= 0 2) P(S) = 1 3) P(E1 U E2 U ...) = P(E1) + P(E2) + ... if all are mutually exclusive Why are the axioms defined in such a way? Why not this simple axiom: - Probability of an event is number of favorable outcomes divided by total number of outcomes?

in this problem we drop the use of parentheses when this step is justified by associative axioms. thus we write $\displaystyle x^2+2x+3\,\,instead\,\,of\,\,\left(x^2+2x\right)+3\,or\,x^2+\left(2x+3\right)$. tell what axioms justify the statement: 1. $\displaystyle...

Let ##G## be a set equipped with a binary associative operation ##\cdot##. In both of the following situations, we have a group: 1) ##G## is not empty, and for all ##a,b\in G##, there exists an ##x,y\in G## such that ##bx=a## and ##yb=a##. 2) There exists a special element ##e\in G##...

I was just wondering, is it possible? It's regarding a debate on whether mathematics is an invention or discovery.

I would like to get some foundational concepts straight about GR as it is currently understood (I guess it is not seen exactly the same it was almost a century ago even if the basic concepts remain, this I would like to elucidate here too). For instance, I understand that a basic axiom of GR as...

Hello, I'd like to make a, probably stupid, question regarding the axioms that define a vetor space. Among them, there are the axioms: λ\cdot(μ\cdotX) = (λμ)\cdotΧ (1) and 1\cdotΧ=Χ (2) for all λ,μ in the field and for all X in the vector space, where 1 is the identity of the...

Is it possible to define sets from just the peano axioms? Usually when people use the peano axioms as the basis of their math they just assume the existence of sets but without axioms regarding sets we technically shouldn't just say they exist. Oh, also there are two versions of the...

Just like the Euler-Lagrange differential equation $$ \frac{\partial{\mathcal{L}}}{\partial{q}} = \frac{d}{dt}\frac{\partial{\mathcal{L}}}{\partial{\dot{q}}} $$ the hamiltonian equations $$ \frac{\partial{H}} {\partial{q}} = -\dot{p} $$ $$ \frac{\partial{H}} {\partial{p}} = \dot{q} $$ and the...

Homework Statement If A and B are events, use the axioms of probability to show that: if B \subset A, then P(B) \leq P(A) Homework Equations Axiom 1: P(n) \geq 0 Axiom 2: P(S)=1 Axiom 3: If A1,A2,... are disjoint sets, then P(\bigcup _{i} A_{i}) = \sum_{i} P(A_{i}) The Attempt at a...

Homework Statement If A and B are events, use the axioms of probability to show: a) If A \subset B, then P(B \cap A^{C}) = P(B) - P(A) b) P(A \cup B) = P(A) + P(B) - P(A \cap B) Homework Equations Axiom 1: P(x)\geq 0 Axiom 2: P(S) = 1, where S is the state space. Axiom 3: If...

i am studying real analysis from terence tao lecture notes for analysis I. http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/ from what i understand , property is just like any other statement. for example P(0.5) is P(0) with the 0s replaced with 0.5 . so the notes says (assumes ?)...

Homework Statement for the 2x2 matrix [a 12;12 b] is it a vector spaceHomework Equations 1. If u and v are objects in V, then u+v is in V 2. u+v = v + u 3. u+(v+w) = (u+v)+w 4. There is an object 0 in V, called a zero vector for V, such that 0+u = u+0 = u for all u in V 5. For each u in V...

I've seen in some probability theory books that the classical definition of probability is a probability measure, it seems fairly trivial but what is the proof for this? Wikipedia gives a very brief one using cardinality of sets. Is there any other way?

Hello, I am looking for more information on the axiomatic treatment of physics. I have found some articles concerning the axioms of quantum mechanics, i.e. the Dirac-Von Neumann axioms. However, I am having a hard time finding anything on the classical version of these axioms. In these...

Homework Statement Let F = {a + b\sqrt[3]{2}:a,b\inQ}. Using the fact that \sqrt[3]{2} is irrational, show that F is not a field. [Hint: What is the inverse of \sqrt[3]{2} under multiplication?] Homework Equations For a field, For all c \in F, there exists c-1 \in F s.t. c*c-1 =1...

Homework Statement prove 2ab<= a^2+b^2 using order axioms Of course use of other basic axioms for real numbers are also okay. Homework Equations axioms for set of real numbers. The Attempt at a Solution The easy way to do this would be just subtract 2ab from both sides, factor...

Every resource I've looked at just lists the axioms but doesn't tell how or why they were arrived at. To what extent are they arbitrary?

The other day our lecturer was going through field axioms, rules of numbers, I guess what you'd call very elementary number theory. In particular, he was explaining what you can and can't do with ∞, and he mentioned that x/∞=0. I guess I'd just like some clarification on this. Should this not...

Hello everyone! I was trying to prove the propositions that follow the addition axioms as a revision, I got a different proof for the following proposition: If $x+y=x+z$ then $y=z$ My proof was the following: $x+y=x+z$, $(-x)+x+y=(-x)+x+z$, $0+y=0+z$, $y=z$ Rudin however, in his book...

Hey all, So I'm just starting a course in linear algebra, but I don't have much experience with proofs. This problem has been giving me some difficulty. So we have a scalar "a" and vector x. V is a linear space, and x is contained in V. I have to show that if ax=0, where 0 is the zero...

If I have an \aleph_0 number of axioms, does that put a limit on the number of theorems I can have about the real numbers. The number of theorems that we could have is countable. I was just wondering what we might be able to say about how much we could know about math.

Homework Statement Prove \neg x \vee x using Hilbert system. Homework Equations The logical axioms. I'm not sure if I should state them, or whether there is a standard set. It seems to me that different sets are used. Anyway, the ones with disjunction in them are: a \rightarrow a \vee b...

I need to prove that every line is contained by at least two planes using only the incidence axioms. This is what I have so far... Conclusions Justifications 1. Let l be any line. Given 2. l has at least two...

Homework Statement V is an arbitrary vector space and v,w,x are part of V such that v + x = w = x Use vector space axioms to prove v = w I've looked at the axioms for an hour and can not get any lead to start this question.