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Math · Advanced Math
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Using these axioms, please prove all parts of #8The set of rational numbers, , is defined to be the smallest set that satisfies all of the following axioms: If a,y E Q then +y EQand ry EQ. (Closure under addition and multiplication) .Addition and multiplication in Q are commutative, associative, and distributive. . 0 EQsuch that for all E Q, z+0-r. (Additive identity) . For all E Q, there exists Esuch that z + (-x) 0. . 1 E Q with 1メ0, such that for all x E Q, we have 1x = x ● For all x E Q, with zメ0, there exists l/T E Q such that z- . For all ,y Eexactly one of the following statements is true: (Additive inverses) (Multiplicative identity (1/x). (Multiplicative inverses) where < y implies 0<y-r . For all z, y, z EQ, if < y and y<z then x < z. (This property and the previous one say that Q is totally ordered) . For all ,y, z E Q, if y< z then y< z. . For all x,y E Q, if x >0 and y >0 then ry >0 8. Use the axioms for Q to prove the following (a) If x +y-z, then y-z. (Hint: start with y 0y and use that (-a)+-0.) (b) If x + y, then y 0. (Make a clever choice for z from part (c) Ifry 0, then y--. (Use part (a) again.) (d) -(-x)-. (You can use part (a) here too.)

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