FIELD AXIOMS

F1: Commutative laws of Addition and Multiplication.

For any a, b Є R, a+ b= b+ a and

a*b= b*a

2

Cont.

F2: Associative Laws of Addition and Multiplication.

For any a, b, c Є R, a+(b+ c)= ( a+ b) + c and

a* (b* c) = ( a* b )* c

3

Cont.

F3: Distributive law of Multiplication

over Addition.

For any a, b, c Є R,a*(b+ c) = a*b+ a*c

4

Cont.

F4: Existence of Identity Elements for Addition and Multiplication.

There are elements 0, 1 Є R, with 0≠1 such that for all a Є R,

a+0 = a and a *1= a

5

Cont.

F5: Existence of Additive Inverse

For each a Є R, there is an element a’ Є R such that:

a+ a’= 0

( the additive inverse a’ of a is the negative of a)

6

Cont.

F6: Existence of Multiplicative Inverse

For each a Є R, with a≠ 0 there is an element a-1 Є R such that a* a-1 = 1

(the multiplicative inverse a-1 of a is the reciprocal of a)

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Ordering Axioms

• O1: There is a non- empty subset R+ of R, with

0 R+, such that, for each real number a, only

one of the following is true.

aЄ R+, or a’Є R+, or a=0

The set R+ is called the set of positive numbers

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Cont.

• O2: For any a, b ЄR+,

a+ bЄ R+ and

a*bЄ R+

Theorems

• T1: If a+z=a, for all a , then z=0(i.e there is only one additive identity)

• T2: If a*u=a, for all a , the u= 1( i.e there is only one multiplicative identity)

• T3: If a+b=0, for all a , b=a'(i.e there is only one additive inverse).

• T4: If a*b=1, for all a , then b=a-1( i.e thre is only one multiplicative inverse.

Cont.

• T5: (any number multiplied by zero is zero)For each a R, a*0=0

• T6: The real number 0 has not multiplicative inverse.

T7- T9

• T7: For any real numbers a and b,

i) (a +b)'= a'+b'

ii) a'*b= (a*b)'= a*b'

iii) a'*b' = a*b.

•T8: If a R+, then a' R- and viceversa.

•Corollary: If a R- then a' R+

•T9: if a R+, and b R-, then a*b R-

•Corollary: If a,b R-, then a*b R+.

Cont.