Prove VzeC, V2|z| > |Re z| + |Im z|Vze C,

Answered: Image /qna-images/answer/2ca0dc6c-69dc-4090-915f-d245601ebe76.jpg

Answered: Prove VZEC, V2|z|>|Re z| +|Im z| + |Im…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.1: Postulates For The Integers (optional)

Problem 31E: 31. Prove that if is positive and is negative, then is negative.

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Similar questions

Let x and y be in Z, not both zero, then x2+y2Z+.
For what values of c is c(1,2,3)=1?
31. Prove that if is positive and is negative, then is negative.
33. Prove that if is positive and is negative, then is negative.
Prove that if m0 and (a,b) exists, then (ma,mb)=m(a,b).
32. Prove that if is positive and is positive, then is positive.
11. (See Exercise 10.) According to Definition 5.29, is defined in by if and only if . Show that if and only if . 10. An ordered field is an ordered integral domain that is also a field. In the quotient field of an ordered integral domain define by . Prove that is a set of positive elements for and hence, that is an ordered field. Definition 5.29 Greater than Let be an ordered integral domain with as the set of positive elements. The relation greater than, denoted by is defined on elements and of by if and only if . The symbol is read “greater than.” Similarly, is read “less than.” We define if and only if. As direct consequences of the definition, we have if and only if and if and only if . The three properties of in definition 5.28 translate at once into the following properties of in . If and then . If and then . For each one and only one of the following statements is true: . The other basic properties of are stated in the next theorem. We prove the first two and leave the proofs of the others as exercises.
25. Prove that if and are integers and, then either or. (Hint: If, then either or, and similarly for. Consider for the various causes.)
To prove a theorem of the form "If P, then Q" by the indirect method, the first line of the proof should read: Suppose that ___________ is true.
Prove that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.

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Prove VZEC, V2|z|>|Re z| +|Im z| + |Im z| Vze C,