Maurice Tutor

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Category > Management Posted 04 Feb 2018 My Price 9.00

proof of Theorem

Fill in the blanks in the following proof of Theorem 2.2(g). Theorem: If x < y="" and="" z="">< 0,="" then="" xz=""> yz.

 

Suppose x < y="" and="" z="">< 0.="" then="" −="" z=""> – 0 by part (f) and − z > 0 by

We have listed the field axioms and the order axioms as properties of the real numbers. But in fact they are of interest in their own right. Any mathematical system that satisfies these 15 axioms is called an ordered field. Thus the real numbers are an example of an ordered field. But there are other examples as well. In particular, the rational numbers Q are also an ordered field. Recall that

where Z = {0, 1, −1, 2, −2, 3, −3,…}. Since the rational numbers are a subset of the reals, the commutative and associative laws and the order axioms are automatically satisfied. Since 0 and 1 are rational numbers, axioms A4 and M4 apply. Since −(m/n) = (−m)/n and (m/n)−1 = n/m, axioms A5 and M5 hold. It remains to show that the sum and product of two rationals are also rational.

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Maurice Tutor
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Status NEW Posted 04 Feb 2018 07:02 PM My Price 9.00

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