Math3333Day02Section3.11Final - Section 11 Ordered Fields The Field Axioms A1 For all x y x y and if x w and y z then x y w z A2 For all x y x y y x A3

Math3333Day02Section3.11Final - Section 11 Ordered Fields...

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Section 11: Ordered Fields
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The Field Axioms: A1. For all , , and if and then . x y x y x w y z x y w z A2. For all , , x y x y y x A3. For all , , , x y z x y z x y z A4. There is a unique real number 0 such that 0 for all . x x x A5. For each there is a unique real number such that 0 x x x x   M1. For all , , , and if and then x y x y x w y z xy wz M2. For all , , x y x y y x M3. For all , , , x y z x y z x y z M4. There is a unique real number 1 such that 1 0 and 1 for all x x x DL. For all , , , x y z x y z x y x z
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The Order Axioms: O1. For all , , exactly one of the following is true: x y x y x y x y O2. For all , , , if and , then x y z x y y z x z O3. For all , , , if , then x y z x y x z y z O4. For all , , , if and 0, then x y z x y z xz yz
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Theorem: Let x , y , and z be real numbers. Then, (a) If , then x z y z x y (b) 0 0 x
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