# Section 3.3: 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 in the structure $(\mathbb{N};\cdot,E)$ we can define the addition relation $\{|m,n\mbox{ in }\mathbb{N}\}$ . Conclude that in this structure $\{0\}$ , the ordering relation $<$ , and the successor relation $\{|n\in\mathbb{N}\}$ are definable. (Remark: This result can be strengthened by replacing the structure $(\mathbb{N};\cdot,E)$ by simply $(\mathbb{N};E)$ . The multiplication relation is definable here, by exploiting one of the laws of exponents: $(d^{a})^{b}=d^{ab}$ .)
$\mathbb{S}^{2}0Ev_{1}\cdot\mathbb{S}^{2}0Ev_{2}=\mathbb{S}^{2}0Ev_{3}$ , and in $(\mathbb{N};\cdot,E,+)$ (in fact, in $(\mathbb{N};+)$ ) we can define $\{0\}$ , $<$ and $S$ (see Exercise 2 of Section 3.2).