Posted by: **magicalrubidium** | June 6, 2008

## Salamat sa Match Coach

This was taken from ematchcoach. Thanks, guys!😀

SET OF COMPLEX NUMBERS

-Set of Real Numbers ( R )

–Set of Rational Numbers ( Q )

(can be represented as a/b wherein a and b are integers and b is non-zero)

(terminating or non-terminating but repeating)

—Set of Integers ( Z or I )

(Q minus the fractions)

—-Set of Whole Numbers ( W )

(Z or I minus the negative)

—–Set of Natural Numbers or Counting Numbers ( N )

( W minus zero)

–Set of Irrational Numbers ( Q’ )

(Everything not part of Q)

(non-terminating but non-repeating)

AXIOMS

Given: a, b, c element of R

A. Field Axioms

- Closure for Addition
- Closure for Multiplication
- Commutative Property of Addition (matter of order)
- Commutative Property of Multiplication (matter of order)
- Associative Property of Addition (matter of grouping)
- Associative Property of Multiplication (matter of grouping)
- Identity Property of Addition (when 0 is added to a number, the number stays the same.)
- Identity Property of Multiplication (when 1 is multiplied to a number, the number stays the same)
- Inverse Property for Addition (an additive inverse is a number’s negative)
- Inverse Property for Multiplication (a multiplicative inverse is a number’s reciprocal, rule doesn’t apply to 0)
- Distributive Property of Multiplication over Addition

B. Equality Axioms

- RPE (Reflexivity): a=a
- SPE (Symmetry): if a=b, then b=a
- TPE (Transitivity): if a=b, b=c, then a=c
- APE ( Addition)if a=b, then a+c=b+c
- MPE (Multiplication)if a=b, then ac=bc

C. Order Axioms

- Trichotomy: either a>b, a=b or a
- TPI (Transitivity): if a>b, b>c, then a>c
- API (Addition): if a>b, then a+c>b+c
- MPI (Multiplication): if a>b, c>0, then ac>bc, but if c=0, then ac=bc, but if c<0,>

D. Completeness Axiom

-Every subset of R that has an upper bound (UB) should have a least upper bound (LUB).

-Similarly, every such subset that has a lower bound (LB) should have a greatest lower bound (GLB).

UB (Upper Bound)- a number greater than or equal to every element of the set.

LUB (Least Upper Bound)- the least number among all the UB’s.

LB (Lower Bound)- a number less than or equal to every element of the set.

GLB (Greatest Lower Bound)- the greatest number among LB’s

Example exercises

1. S={-3, 0, 3}

2. T={1, ½, ¼, 1/8, 1/16}

3. Set of nonnegative integers

4. S={x|x element of I, x<5/2}

Answers

1. GLB: -3 LUB:3

2. GLB: 1/16 LUB: 1

3. GLB: 0 LUB: none

4. GLB: none LUB: 2

Note: Infinity is not a number.

– Cam😀

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