EDB — 22P

Exercises

1. [22P](Proposed on 2023-01-17) Given two ordered non-empty sets \((X,≤_ X)\) and \((Y,≤_ Y)\), and \(f:X→ Y\) as defined in [07V]↺↻.

   • If \(A⊆ X\) and \(m=\max A\) then \(f(m)=\max f(A)\); similarly for the minimums;

   • \((X,≤_ X)\) is totally ordered if and only if \((Y,≤_ Y)\) is;

   • \((X,≤_ X)\) is well ordered if and only if \((Y,≤_ Y)\) is.

   • Suppose that \((X,≤_ X)\) and \((Y,≤_ Y)\) are well ordered, let \(S_ X\) and respectively \(S_ Y\) be the functions ”successor”, [1Z0]↺↻, then we have that \(x\) is not the maximum of \(X\) if and only if \(f(x)\) is not the maximum of \(Y\), and in this case \(y=S_ X(x)\) if and only if \(f(y) = S_ Y(f(x))\).