#bounded sets

[kgdrendel]

Thinking Outside the Circle and Focusing on the Center: What Direction are You Moving?

If we are not challenged to rethink what we think we know from time to time, we are not likely coming into close enough contact with Jesus.

A paradigm shift inspired by mathematics and Jesus.

[kgdrendel]

Thinking Outside the Circle and Focusing on the Center: What Direction are You Moving?

Thinking Outside the Circle and Focusing on the Center: What Direction are You Moving?

I watched the Chapelstreet church service today and listened to the sermon by Jeff Frazier in Batavia, IL. It was just what I needed to hear. Not that it tills new ground; it covers familiar ground from a new angle. It avoids the ruts of old, tired ways of thinking and finds fresh new ground (for me) from which to approach how we see Jesus. The sermon today was inspired by Matt. 9:9-13.[i] You…

View On WordPress

[fardell24b]

Math 101 revision - Lecture 9

10 March 2000

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

   '�|�QT