#bounded sets

Math 101 – Lecture 8

Basic Arithmetic

The set of real numbers R has the following properties a+b = b+a R for a, b R  a+ 0 = 0 + a, the additive identity.

For each a ( 0) R, there exists b R such that ab = 1 where 1 is the multiplicative identity a*1 = 1*a = a.

Order for any two real numbers a, b R  only one of the following is true:

 a > b or

a = b or

a < b

 If a, b, c R and a > c, c > b then a > b

           If a, b, c R and a > b then a + c > b + c

If a, b, c R and a > b, if c > 0 then ac > bc

 From these properties, together with the deankind axiom all the familiar results for real numbers.

Eg Index Laws a^m a . a . a .... a, m IN

                       {          m times  }

           Extend to a^-m = 1/(am) = 1(a . a . a .... a) =               1/a . 1/a . 1/a .... 1/a

                                                           {          m times  }        {          m times  }

           Extend to a^(p/q), p, q Z

Eventually extend to a^-σ , σ R

 Absolute value – a ‘distance measure’ on R

|a| = {a, a 0

           -a, for a < 0

Theorem (properties of absolute values)

1.      |a| 0 for all a R, note  |a| = 0 if any only if a = 0

2.      |ab| = |a| * |b|, for all a, b R

Cases

(i)                 |ab| = ab = |a| |b|     ie ab > 0 for [(i) a, b > 0

(ii) a, b < 0

(ii)               |ab| = ab = (-a)(-b)

3.      |a|^2 = a^2

4.      |a+b| |a| + |b|, a, b R

Proof: If a+b<0 then |a + b| = -(a + b)

                                               = (-a) b),

Now – a |a|, for any a R

 If a + b > 0 then |a + b| = a + b

But a |a| and b |b|

|a + b| |a| + |b|

5.      |a – b| |a| - |b|, for any a, b  R

-||-

Bounded Sets

Infinite sets : 1) Q

                 2) Prime numbers

                 3) Z

     4) {x  Q: -1 x 1}

4) is a subset of an interval of the real number line.

[a, b] = {x^(R): a x b} – a closed interval.

[a, b] = { x^(R): a x b }

Also have [a, b] and [a, b]

 Bounded sets: - A set S R is bounded above if thre is K R, such that K for all x  S

     S is said to be bounded below.

                 If there is k  R such that x  k for all x    S

                       S is bounded if it bounded above and below.

*The least upper bound (supremum) for a bounded set S R is K R such that K x for some x    S and any > 0

*The greatest lower bound (infinum) for is K R such that [ k x, x S

                                                                                                     K + > x for some x S and any > 0