# Section 3.2: Problem 1 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
Show that any eventually periodic set of natural numbers is definable in the structure $\mathfrak{N}_{A}$ .
Every finite subset and its complement in $\mathbb{N}$ can be defined in $\mathfrak{N}_{S}$ (Exercise 4 of Section 3.1), and, hence, in $\mathfrak{N}_{A}$ . Therefore, by using disjunctions and conjunctions we can modify any finite number of points of a subset, and it is sufficient to show that every periodic set of natural numbers is definable in $\mathfrak{N}_{A}$ . But every periodic set $D$ with period $p$ is determined by memberships of the first $p$ natural numbers. Hence, if $D\cap\{0,\ldots,p−1\}=\{d_{1},\ldots,d_{k}\}$ , then $D$ is defined by $\exists x(v_{1}=d_{1}+\underbrace{x+\ldots+x}_{p}\vee\ldots\vee v_{1}=d_{k}+\underbrace{x+\ldots+x}_{p})$ .