Suppose z=a+b i, w=c+d i. Define z<w if a<c, and also if a=c...

Suppose $z=a+b i, w=c+d i$. Define $z<w$ if $a<c$, and also if $a=c$ but $b<d$. Prove that this turns the set of all complex numbers into an ordered set. (This type of order relation is called a dictionary order, or lexicographic order, for obvious reasons.) Does this ordered set have the least-upper-bound property?

Key Concepts

Ordered Set

An ordered set is a set equipped with a relation that arranges its elements in a sequence according to certain criteria. For the relation to be a valid ordering, it must satisfy properties like transitivity, antisymmetry, and comparability (totality), meaning any two elements can be compared. This forms the basis for understanding how elements can be 'less than' or 'greater than' each other in a mathematically rigorous way.

Lexicographic (Dictionary) Order

Lexicographic order is a method of ordering pairs or sequences where the first elements are compared first, and if they are equal, the subsequent elements are compared to decide the order. In the context of complex numbers, this means comparing the real parts first, and if they are equal, comparing the imaginary parts. This ordering mimics the way words are sorted in a dictionary and provides a way to impose a total order even on sets that do not naturally have one.

Least-Upper-Bound Property

The least-upper-bound (or supremum) property is a completeness criterion for ordered sets. It states that every non-empty subset that is bounded above must have a smallest element among all its upper bounds, called the supremum. While this property holds for the set of real numbers, it does not necessarily hold for all ordered sets, particularly those ordered lexicographically, which can exhibit gaps where a supremum does not exist within the set.

Total Order

A total order is a specific type of order relation where every pair of elements in the set is comparable; that is, for any elements x and y, either x is less than y, y is less than x, or they are equal. This property is essential in the definition of lexicographic order for complex numbers, ensuring that every pair of complex numbers can be ordered, even though the resulting set might lack other properties like completeness.

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