Well Ordered Sets: Disjoint Union w/ R, S, A x B

[Kontilera]

Homework Statement

Show that for two well ordered sets, (A, R) and (B, S), the disjoint union of A and B will be well ordered by the relation $$R \cup S \cup A \times B$$.

The Attempt at a Solution

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I honesly don't know how to start at this one..

 

[Vargo]

Well let's start by describing the ordering that they suggest.

An ordering is a relation which is a set of ordered pairs. If (x,y) belongs to the relation then x<y. So we have the set of ordered pairs R union S union AxB. AxB is the set (a,b) where a is from A and b is from B. By this ordering all elements of A are less than all elements of B.

So given any two elements of A union B, can you compare them? If so, then this is a total ordering.

Given any subset of A union B, is there a least element? Think of it in terms of the number line. All the elements of A are to the left of all the elements of B. And A and B are themselves well ordered.

Hope that helps.