All Questions
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Using field axioms to prove the next
how can it be proved using field axioms that
$\frac{1}{\sqrt[3]{100}}=\frac{\sqrt[3]{10}}{10}$
I have the next sketch proof:
First I applied the definition of quotient. Then I used that $1=(\sqrt[3]{...
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In the book by Apostol "calculus volume 1" how to prove that sum of two integers is an integer?
In Apostol's book we start by defining a set called the set of real numbers which satisfies the field and order axioms. Then we define the set of positive integers as being the subset of every ...
user907912
2
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2
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Is it really important to do axiomatic study of real numbers before learning Calculus? [closed]
I am currently beginning with Calculus Volume 1 by Tom M. Apostol . It has an introduction chapter divided into 4 parts namely
Historical introduction
Basic concept of set theory
A set of axioms ...
- 191
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1
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173
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How to draw Axiom of Continuity : $\exists c \in\mathbb{R} :\forall a \in A, \forall b \in B \implies a \leq c \leq b$
In Real Analysis, while we are constructing the Real Numbers Axiomatically, we (in some books) define one important Axiom, Axiom of Continuity, which goes like this :
"If $A, B\subseteq\mathbb{R}$...
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1
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149
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Why $x=x$? Why $x=y$ and $y=z$ imply $x=z$? (Assume, $x$, $y$ and $z$ are reals.)
I recently have been studying the axiomatic construction of the set of real numbers through the Peano axioms for the natural numbers. It seems to me, the only things needed to proceed with this body ...
1
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1
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242
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Continuity axioms and completness axioms for real numbers are the same things?
Sometime I read that Dedekind's axiom is a continuity axiom, and sometimes I read that it's a completeness axiom. Besides Dedekind's axiom is equivalent to other properties as I read here in The Main ...
4
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1
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The axiomatic method to real number system VS the constructive method(genetic method)
According to book Georg Cantor: His Mathematics and Philosophy of the Infinite -
Joseph Warren Dauben , David Hilbert claimed that the axiomatic method to real number system is more secure than the ...
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Proof of |x · y| = |x| · |y| using axioms of real numbers
I'm trying to prove |x · y| = |x| · |y| using only the axioms of real numbers. I'm using the definition of the modulus function to be below. I thought I should start by distinguishing four cases like (...
user378721