Weekly Math 4 (Tacuboy): Week 1 (First Quarter)

NOTE: Ang Math 4 (Tacuboy) ay ang ating Thursday segment

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

1. Closure for Addition
2. Closure for Multiplication
3. Commutative Property of Addition (matter of order)
4. Commutative Property of Multiplication (matter of order)
5. Associative Property of Addition (matter of grouping)
6. Associative Property of Multiplication (matter of grouping)
7. Identity Property of Addition (when 0 is added to a number, the number stays the same.)
8. Identity Property of Multiplication (when 1 is multiplied to a number, the number stays the same)
9. Inverse Property for Addition (an additive inverse is a number’s negative)
10. Inverse Property for Multiplication (a multiplicative inverse is a number’s reciprocal, rule doesn’t apply to 0)
11. Distributive Property of Multiplication over Addition

B. Equality Axioms

1. RPE (Reflexivity): a=a
2. SPE (Symmetry): if a=b, then b=a
3. TPE (Transitivity): if a=b, b=c, then a=c
4. APE ( Addition)if a=b, then a+c=b+c
5. MPE (Multiplication)if a=b, then ac=bc

C. Order Axioms

1. Trichotomy: either a>b, a=b or a
2. TPI (Transitivity): if a>b, b>c, then a>c
3. API (Addition): if a>b, then a+c>b+c
4. 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.

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