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|?

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On 04 Sep 2007, 20:30.