Chapter 9.6, Problem 54E

ermine whether each of these posets is well-ordered.

a) $\left(S,\le \right)$, whereS= {10, 11, 12,…}b) $\left(Q\cap \left[0,1\right],\le \right)$(the set of rational numbers between o and1inclusive)

c) $\left(S,\le \right)$,where Sis the set of positive rational numbers with denominators not exceeding 3

d) $\left(Z^{-},\ge \right)$,where $Z^{-}$is the set of negative integers

A poset (R, $\preccurlyeq$)is well-founded if there is no infinite decreasing sequence of elements in the poset, that is, elementsx₁,x₂, ...,x_nsuch that $\cdot \cdot \cdot \prec x_n\prec \cdot \cdot \cdot \prec x_2\prec x_1$. A poset (R, $\preccurlyeq$) is dense if for all

$x\in S$and $y\in S$with $x\prec y$, there is an element $z\in R$such that $x\prec z\prec y$.