How to define a well-order in C(set of complex numbers)? ord(C)?

This is a question from Set Theory which was on my last exam.

A few definitions to help:

Definition: Partial order in a set A is a relation R in A which is Reflexive, Anti-symmetric, Transitive.

Definition: Let A be an orderd set with the property that every subset of A contains a first element. A is called a well-ordered set.

Well-oredering theorem: Every set can be well orderd.

How to define a well-order in C(set of complex numbers)? ord(C)?visual c++ express

Here's a total ordering for all of C:

Every element of C is of the form a + bi.

for two complex numbers c, d if |c| > |d| then c > d, where |a+bi| = a^2 + b^2

If |c| = |d| let c = a + bi, d = f + gi

then when a > f we let c > d

If a = f, then either b = g, in which case they are equal, or b = -g, in which case choose the one with the positive part to be bigger.

How to define a well-order in C(set of complex numbers)? ord(C)?c++ tutorial

This is a total ordering, but it is not a well-ordering. There are many sets with no smallest element in this ordering. For example, {1/n:n in N} Report It |||But first you need an order on C and you can do this lexicographically. So the problem of well ordering C reduces to well ordering the set of real numbers R.

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How to define a well-order in C(set of complex numbers)? ord(C)?

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