MthT 430 Projects Chapter 1a
MthT 430 Projects Chapter 1a
page 2 In class September 5, 2007






The Triangle Inequality and Applications
For the time being, assume (P1) - (P12), and
|a| = ì
ï
í
ï
î
a,
a ³ 0
-a,
a £ 0.


The Triangle Inequality says that
|a + b| £ |a| + |b|.
1.
Show that
|a - b| £ |a| + |b|.


2.
Show that
|a b| = |a|·|b|.


3.
Show that
|a | £ |a - b| + |b|.


4.
Show that
||a| - |b| | £ |a - b|.
5.
List all numbers such that |a| = 0.


6.
Show that if
0 < a < b,
then
0 < b-1 < a-1.


Work on Chapter 1, Problems 20 and 21 in Spivak
20.
Prove that if
|x - x0| < e/2 and |y - y0| < e/2,
then
|(x + y) - (x0 + y0)| < e.


OE
Let the set of numbers OE consist of the two objects
{odd, even}
Here is the addition table:

    + (plus)         odd         even    

    odd         even         odd    

    even         odd         even    

Here is the multiplication table:

    · (times)         odd         even    

    odd         odd         even    

    even         even         even    



This set of Numbers satisfies (P1) - (P9).


1.
Which element has the role of 0?


2.
Which element has the role of 1?


3.
Is it possible to define a set of positive numbers P such that (P10) - (P12) are satisfied?


4.
Is it possible to define an absolute value on OE with all of the properties:
A1
For all a in OE, |a| is a real number, |a| ³ 0,
A2
|a| = 0 iff a = 0,
A3
|a + b| £ |a| + |b|,
A4
|a ·b| = |a| ·|b|?



File translated from TEX by TTH, version 3.77.
On 04 Sep 2007, 20:30.