« Section 1.4: Problem 2 Solution

# Section 1.4: Problem 3 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
We can generalize the discussion in this section by requiring of $\mathcal{F}$ only that it be a class of relations on $U$ . $C_{\star}$ is defined as before, except that $x_{0},x_{1},\ldots,x_{n}$ is now a construction sequence provided that for each $i\le n$ we have either $x_{i}\in B$ or $\in R$ for some $R\in\mathcal{F}$ and some $j_{1},\ldots,j_{k}$ all less than $i$ . Give the correct definition of $C^{\star}$ and show that $C^{\star}=C_{\star}$ .
$C^{\star}$ is the intersection of all inductive sets $S$ such that $B\subseteq S$ and if $x_{0},x_{1},\ldots,x_{n-1}\in S$ and $\in R$ for some , then $x_{n}\in S$ .
$C_{\star}$ is inductive, therefore, $C^{\star}\subseteq C_{\star}$ .
And for any construction sequence $$ , by ordinary induction, each $x_{j}\in C^{\star}$ , therefore, $C_{\star}\subseteq C^{\star}$ .