All Questions
Tagged with real-numbers axioms
76
questions
4
votes
1
answer
322
views
The axiom of regularity and the real numbers
I'm having trouble understanding how there can be sufficiently many distinct elements for $\mathbb{R}$ to exist with its properties and yet still be a set.
(The ...
- 151
0
votes
3
answers
3k
views
Using only the field axioms of real numbers prove that $(-1)(-1) = 1$
Using only the field axioms of real numbers prove that $(-1)(-1) = 1$
(1) I start with an obvious fact:$$0 = 0$$
(2) Add $(-1)$ to both sides of the equation:
$$0 + (-1) = 0+ (-1)$$
(3) Zero is the ...
- 3,709
1
vote
3
answers
272
views
Prove, using only the field axioms of real numbers, that $0/0$ is undefined.
Prove, using only the field axioms of real numbers, that $0/0$ is undefined. I have thought about it for a while and come up with an idea how to solve this. First, I would like to prove (using field ...
- 3,709
-1
votes
1
answer
5k
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Using only the field axioms of real numbers, prove that $-x = (-1)x$ [duplicate]
Using only the field axioms of real numbers, prove that $-x = (-1)x$
Ths is how I attempted to solve this problem:
$$1+(-1)=0 \iff x(1+(-1))=0\cdot x \iff x+(-1)x=0\iff(-1)x=-x$$
However, I am not ...
- 3,709
0
votes
3
answers
271
views
Using the axioms of real numbers prove that 0 < 1 [closed]
These are the axioms that I am allowed to use:
(1) $x + 0 = 0 + x = x$ (2)$x \cdot 1 = 1 \cdot x = x$
(3) $xy = 1 \iff y = \frac{1}{x}$, $x \neq0$
(4) $x+y = 0 \iff y = -x$
(5) $ x(y+z) = xy + xz$ ...
- 3,709
9
votes
2
answers
838
views
axioms of real numbers without multiplication
Consider the axioms of real numbers https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach
and suppose we remove the multiplication operation and its properties. Do we loose something?
I have ...
- 21.6k
0
votes
1
answer
73
views
How to prove this theorem using field axioms of real numbers?
I need to prove that $0*a=a*0=0$
I'm guessing I have to use the existence of inverse axiom.
But I'm not sure how the proof goes.
Thanks a lot in advance!
- 415
0
votes
2
answers
2k
views
How to prove the theorems of algebra using axioms?
Like the vast majority of math students I'm having trouble with proofs. I would like to get some advice on how to approach proofs. What are the steps that need to be done?, how do I know that the ...
- 415
0
votes
1
answer
176
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Axiomatics of Real Numbers - should $0\neq 1$ be considered as axiom?
I am analysing axiomatic approach to defining real numbers. There are two axioms that postulate existence of $0$ and $1$, namely (according to my notes):
There exists an element $0\in\mathbb{R}$ such ...
- 908
2
votes
4
answers
532
views
Isn't it obvious that $0$ and $1$ are distinct?
The field axioms for the real number system contain the following statements concerning the existence of neutral or identity elements for addition and multiplication:
(1) There exists a real number,...
user372009
1
vote
1
answer
105
views
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
0
votes
1
answer
109
views
Proving some basic algebraic statements using the axioms of the real numbers
I am trying to rigorously understand how algebra works by deriving everything from the axioms of the real numbers. I thought this wouldn't be too difficult but it seems I don't have any idea where to ...
- 777
7
votes
0
answers
701
views
Geometric basis for the real numbers
I am aware of the standard method of summoning the real numbers into existence -- by considering limits of convergent sequences of quotients.
But I never actually think of real numbers in this way. I ...
- 2,146
1
vote
1
answer
24
views
If $x+y=(x_1y_1, ..., x_ny_n)$ and $c\cdot '\ x=x^c_1, ..., x^c_n$, how to show that with these two operation $V$ is a subspace?
Let $V=(R^+)^n=\{(x_1, ..., x_n)| x_i\in R^+$for each $i\}$. In $V$ define a vector sum operation $+'$ by $x+y=(x_1y_1, ..., x_ny_n)$ and scalar multiplication $\cdot '$ by $c\cdot '\ x=x^c_1, ..., x^...
- 871
4
votes
2
answers
1k
views
Axiomatic definition of the real numbers and uncountability
There are several approaches in defining the real numbers axiomatically and suppose that we have some set of axioms $A$ which completely characterize rational numbers and which does not mention ...
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