All Questions
Tagged with real-numbers axioms
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Tarski axioms of real numbers
How does the Tarski axioms of real numbers imply that for each x,y,z ( x<y if and only if x+z < y+z ) ?
By using the 1st and 6th axioms it's easy to demonstrate that x+z<y+z implies x<y. ...
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For any numbers $a, b,$ and $c,$ $a + b = a + c$ if and only if $b = c$ [duplicate]
I was reading about the field of real numbers $\mathbb{R},$ and a basic question arose in my mind.
How one should prove that, for any numbers $a, b,$ and $c,$ $a + b = a + c$ if and only if $b = c?$
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Help with a proof of a consequence from the axioms of addition and multiplication
While reading through Analysis 1 by Vladimir A. Zorich, I encountered this proof which has this 1 step I can't understand. Here is the consequence and the proof:
For every $x\in \mathbb R$ the ...
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Construction of Real Number by Dedekind Cuts [closed]
I was studying Axiomatic Set Theory, and I have 2 questions about the construction of real numbers using Dedekind cut:
We define a real number using the Dedekind cut: $x_{\mathbb{R}} = \left \{ p \...
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Is it possible to create the smallest real positive number by axiome?
I know that with standard math there is no "smallest positive real number". But, the same way we created Aleph Null by axiome, can we create the axiome below?
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Are Real Numbers a Formal System?
I don't know a lot of mathematics but I have noticed that every branch of Mathematics has the same structure: some axioms (For example in Geometry might be Euclid's Axioms, in Probability might be ...
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How can we show that if $|x| \le 1/n$ for all natural numbers, n, then $x = 0$?
I was thinking about how to define the real number system axiomatically, and can't find anywhere a proof that $$\left[\forall n \in \mathbb{N}\left(|x| \le \frac{1}{n}\right)\right] \Rightarrow [x = 0]...
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Prove using the axioms that the square of any number is nonnegative
How do you prove $\forall x\in \Bbb{R}, x^2 \ge 0$ using the axioms?
My lecturer hinted you should split the cases up into $x=0$ and $x \ne 0$.
The $x=0$ case seems pretty obvious: $x^2 =x \cdot ...
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Prove using the axioms that $x>0$ implies $-x<0$
How to prove equations that if $x>0$, then $-x<0$ using the axioms of the real numbers $\Bbb{R}$ (if $x \in \Bbb{R}$)?
My university lecturer gave this as an exercise and I am stuck on which ...
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Question of proof of archimedean property
For every real number x there exists an integer $n$ such that $n>x$.
The book is using contradiction,
Suppose $x$ is a real number such that $n≤x$ for every $n$,that mean $x$ is the upper bound ...
user743730
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Proof that $\frac{1}{2} + \frac{1}{2} = 1$ using just the algebraic properties of $\mathbb R$
Like the title says, can you prove rigorously that $\frac{1}{2} + \frac{1}{2} = 1$ using only the nine field properties of $\mathbb R$? I don't know if addition and multiplication are supposed to be ...
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Why is a single nonnegative number smaller than a sum of nonnegative numbers?
I know this sounds like an incredibly dumb question, but why is a single nonnegative number smaller than a sum of nonnegative numbers in a vector? I know it's true, but I want to know why it's true. ...
- 557
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How to derive the axiom no. 15 from the Cantor's and Archimedean axiom?
How could one substitute the (15th) axiom of completeness with Archimedean and Cantor's axiom?
We discussed Cantor's axiom as well as Archimedean in analysis lectures and were told this question might ...
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How can I prove this statement without using reduction to absurdity?
$\forall a,b\in\mathbb R[\forall c\in \mathbb R(c>a\implies c>b)\implies a\ge b]$
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Are there an axioms for cartesian space?
Do we have any axioms that allow us represent Cartesian coordinates on a graph in euclidean space or is it purely intuitive? It's easy to intuitively justify where $(0,1)$ and $(1,1)$ would lie in ...
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