EDB — 08F

Exercises

1. [08F]Prerequisites:[01P]↺↻, [07Z]↺↻, [084]↺↻, [080]↺↻, [08C]↺↻.

   Let be given two well-ordered non-empty sets \((X,≤_ X)\) and \((Y,≤_ Y)\). Show that

   1. there is an initial segment \(S\) of \(X\) and a strictly increasing monotonic bijective function \(g:S→ Y\); or ¹

   2. there is an initial segment \(T\) of \(Y\) and a bijective strictly increasing monotonic function \(g:X→ T\).

   In the first case we will write that \((Y,≤_ Y)⪯ (X,≤_ X)\), in the second that \((X,≤_ X)⪯ (Y,≤_ Y)\). (Note that in the first case you have \(|Y|≤ |X|\) and in the second \(|X|≤ |Y|\)). By the previous exercise, the map \(g\) and its segment are unique.

   Solution 1

   [08G]↺↻