2.3 Algebraic and Ordering Properties of R
Assumptions A1-A18 will be used to derive basic and familiar algebraic and ordering properties of the real numbers. The theorems and examples that follow are designed to do two things. First, they will give a feel for how to write proofs of this sort. Second, they will serve as assumptions for the exercises and theorems from later sections.
These are the (A1 – A18):
(A1) Properties of equality;
(A2) Addition is well defined;
(A3) Closure property of addition;
(A4) Associative property of addition;
(A5) Commutative property of addition;
(A6) Existence of an additive identity;
(A7) Existence of additive inverses;
(A8) Multiplication is well defined;
(A9) Closure property of multiplication;
(A10) Associative property of multiplication;
(A11) Commutative property of multiplication;
(A12) Existence of a multiplicative identity;
(A13) Existence of multiplicative inverses;
(A14) Distributive property of multiplication over addition;
(A15) 1 ≠ 0
(A16) Trichotomy law;
(A17) If a > 0 and b > 0, then a + b > 0.
(A18) If a > 0 and b > 0, then a∙b > 0.
As the previous exercises, my friend (Ahmad Wachidul Kohar) and I try to make a resume and solve the exercises in this section which can be downloaded in this following link: 2.3 Algebraic and Ordering Properti of R