Exercises
[08F]Prerequisites:[01P], [07Z], [084], [080], [08C].
Let be given two well-ordered non-empty sets \((X,≤_ X)\) and \((Y,≤_ Y)\). Show that
there is an initial segment \(S\) of \(X\) and a strictly increasing monotonic bijective function \(g:S→ Y\); or 1
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.
1