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Unformatted text preview: Section 1.3: Properties of Real Numbers In this section, we take an axiomatic approach to the real numbers, that is, we assume that R exists and satisfies all of the properties of a complete ordered field , rather than proving that this is the case. Definition: There exists a set R containing the rational numbers Q , which is a complete ordered field , a system endowed with two binary operations addition and multiplication, usually denoted + and g respectively, and an order relation, denoted <, such that the following properties hold: 1. For every , x y ∈ R , x y + ∈ R ; that is, R is closed under addition. 2. For every , x y ∈ R , x y ∈ R g ; that is, R is closed under multiplication. For convenience, we will usually write x y g as simply xy . 3. For every , x y ∈ R , x y y x + = + and xy yx = ; that is, addition and multiplication are commutative. 4. For every , , x y z ∈ R , ( ) ( ) x y z x y z + + = + + and ( ) ( ) x yz xy z = ; that is, addition and multiplication are associative. 5. There exists elements 0,1 ∈ R such that for every x ∈ R , x x + = and 1 x x = g ; that is, there exists additive and multiplicative identities. Moreover, it can be shown that these identities are unique. 6. For every x ∈ R , there exists y ∈ R such that x y + = . Similarly, for every x ∈ R with x ≠ , there exists y ∈ R such that 1 xy = . 7. For every , , x y z ∈ R , ( ) x y z xy xz + = + ; that is, distribution holds in R 8. For every , , x y z ∈ R , x y < implies x z y z + < + 9. For every , , x y z ∈ R , x y < and y z < implies x z < 10. For every , x y ∈ R , one and only one of the following hold: x y < , x y = , or y x < . 11. For every , , x y z ∈ R , if x y < and z < , xz yz < . 12. Every nonempty subset of R that is bounded above has a least upper bound in R ....
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- Spring '08
- Real Numbers, upper bound, β, α