EDB — 22P

Exercises

1. [22P](Proposed on 2023-01-17) Given two ordered non-empty sets $(X,{\le }_X)$ and $(Y,{\le }_Y)$, and $f:X\to Y$ as defined in [07V]↺↻.

   • If $A\subseteq X$ and $m=maxA$ then $f(m)=maxf(A)$; similarly for the minimums;

   • $(X,{\le }_X)$ is totally ordered if and only if $(Y,{\le }_Y)$ is;

   • $(X,{\le }_X)$ is well ordered if and only if $(Y,{\le }_Y)$ is.

   • Suppose that $(X,{\le }_X)$ and $(Y,{\le }_Y)$ 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 $f(y)=S_Y(f(x))$.