The Student Room Group
Uni students - Complete our survey >>
Uni students - Complete our survey >>
Uni students - Complete our survey >>

Deriving statements using the axioms of real numbers

I've been asked to derive the following using the axioms of the real numbers:

0<x<y⇒1y<1x0<x<y \Rightarrow \frac{1}{y} <\frac{1}{x}

0<x<y,0<z<w⇒xz<yw0<x<y , 0<z<w \Rightarrow xz<yw

x<0,y<0⇒0<xyx<0 , y<0 \Rightarrow 0<xy

I've managed to get through proving ones like x+x=2x x+x=2x but the inequalities are confusing me. I'd simply like a starting point for each and any explanations as to how I can improve my thought processes, i.e. help beginning to "think like a mathematician".

The Axioms I've been given are below

Axioms.png
(edited 7 years ago)
LectureNotesCalculus1-6.pdf0.2KB
You need to post the axioms you've been given for the real numbers (there isn't one "standard set" of axioms AFAIK).
Reply 2
Avatar for TajwarC
TajwarC
OP
17
Original post by DFranklin
You need to post the axioms you've been given for the real numbers (there isn't one "standard set" of axioms AFAIK).


Ah, wasn't aware of this, I've updated the OP
Original post by TajwarC
Ah, wasn't aware of this, I've updated the OP
Unfortunately, that PDF doesn't open (for me, at any rate).
Reply 4
Avatar for TajwarC
TajwarC
OP
17
Original post by DFranklin
Unfortunately, that PDF doesn't open (for me, at any rate).


Apologies, I've uploaded a screenshot into the OP now
Original post by TajwarC
I've been asked to derive the following using the axioms of the real numbers:

0<x<y⇒1y<1x0<x<y \Rightarrow \frac{1}{y} <\frac{1}{x}

0<x<y,0<z<w⇒xz<yw0<x<y , 0<z<w \Rightarrow xz<yw

x<0,y<0⇒0<xyx<0 , y<0 \Rightarrow 0<xy

I've managed to get through proving ones like x+x=2x x+x=2x but the inequalities are confusing me. I'd simply like a starting point for each and any explanations as to how I can improve my thought processes, i.e. help beginning to "think like a mathematician".


Given your axioms, I think that you need to prove quite a number of intermediate results. You may already have done some of this in lectures, of course.

For example for the final part, you could show that:

1) x<0⇒−x>0x < 0 \Rightarrow -x > 0
2) x>0,y>0⇒0<xyx > 0, y > 0 \Rightarrow 0 < xy
3) (−x)(−y)=xy(-x)(-y) = xy

or something along those lines.
Reply 6
Avatar for TajwarC
TajwarC
OP
17
Original post by atsruser
Given your axioms, I think that you need to prove quite a number of intermediate results. You may already have done some of this in lectures, of course.

For example for the final part, you could show that:

1) x<0⇒−x>0x < 0 \Rightarrow -x > 0
2) x>0,y>0⇒0<xyx > 0, y > 0 \Rightarrow 0 < xy
3) (−x)(−y)=xy(-x)(-y) = xy

or something along those lines.


Ah I'm seeing why you've considered −x-x and −y-y. Should be enough to get me going for this one, thanks.
Original post by TajwarC
Ah I'm seeing why you've considered −x-x and −y-y. Should be enough to get me going for this one, thanks.


My pleasure.

You may find it useful to prove the following for part 1):

x>0⇒1x>0x > 0 \Rightarrow \frac{1}{x} > 0

(Use trichotomy and contradiction)

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you