Axioms - Axiom for real numbers PDF

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Axiom for real numbers...

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MATH 3000, Winter 2019 Axioms for the real numbers

Field axioms There exist binary operations + and · (addition and multiplication) such that 1. (Addition is associative) If x, y, z ∈ R, then (x + y) + z = x + (y + z). 2. (Addition is commutative) If x, y ∈ R, then x + y = y + x. 3. (Additive identity) There exists a number 0 ∈ R such that 0+x= x for any x ∈ R. 4. (Additive inverse) For each x ∈ R, there exists a number y ∈ R such that x + y = 0. 5. (Multiplication is associative) If x, y, z ∈ R, then (xy )z = x(yz ). 6. (Multiplication is commutative) If x, y ∈ R, then xy = yx. 7. (Multiplicative identity) There exists a number 1 ∈ R, with 1 6= 0, such that 1·x= x for each x ∈ R. 8. (Multiplicative inverse) If x ∈ R and x 6= 0, then there exists y ∈ R such that xy = 1. 9. (Distributive law): If x, y, z ∈ R, then x(y + z) = xy + xz.

Order axiom 10. There exists a subset P ⊆ R such that (i) If x, y ∈ P , then x + y ∈ P and xy ∈ P (ii) If x ∈ R exactly one of the following statements is true: x ∈ P,

x = 0,

−x ∈ P.

Least upper bound axiom 11. Every nonempty subset of R that is bounded above has a least upper bound....