# Determine whether or not the poset $(\mathbb{N}, \propto)$ has a least element

Define a relation $$\propto$$ on the natural numbers $$\mathbb{N}$$ by declaring that for $$x, y \in \mathbb{N}$$, $$x \propto y \iff (x=y)$$ or $$(3x \leq y)$$

a) Show that $$\propto$$ is a partial order on $$\mathbb{N}$$

b) Determine whether or not the subset $$\{2, 6, 8\}$$ of $$\mathbb{N}$$ is totally ordered with respect to $$\propto$$. Explain your answer.

c) Determine whether or not the poset $$(\mathbb{N}, \propto)$$ has a least element.

For part a), I showed that it was reflexive, transitive and antisymetric.

For part b) I determined that the subset was NOT totally ordered because 6 $$\not \propto 8$$ and $$8 \not \propto 6$$

I'm unsure about part c). My guess is that there is no least element because $$1 \not \propto 2$$ so $$1$$ cannot be the least element. However $$\forall n \in \mathbb{N}$$ with $$n>1$$, $$n \not \propto 1$$, so there is no number $$n \not = 1$$ which is the least element.

But I'm sure I'm phrasing this really badly...

• If I understand the definitions correctly, $0$ is a least element, since $3\cdot 0\le n$ for every $n\in\mathbb N$. – hmakholm left over Monica Jun 19 '16 at 1:59
• Ah yes that would be true - but my lectures told me that for this particular course we are not counting zero as a part of the natural numbers (apparently 0 $\in \mathbb{N}$ under some definitions but not others) – maths123 Jun 19 '16 at 2:08

Suppose $$n \ne 1$$ is a least element under $$\propto$$. Since you assume $$\Bbb N = \Bbb Z^+$$, then $$n \ge 2$$. Then $$n \propto 1$$ in particular. However, $$n \ne 1$$, and since $$n \ge 2$$, then $$3n \ge 6$$; thus $$3n \not \le 1$$ and $$n \not \propto 1$$
So the only remaining candidate is $$n=1$$. However, $$1 \not \propto 2$$ because $$3 \cdot 1 \not =3\le 2$$.
Thus, all possible candidates have been removed, and $$\propto$$ has no least element in $$\Bbb N$$. (Though if you were to include zero, then $$0$$ would be the least element, since $$0 \cdot x = 0 \le y$$ for all $$y \in \Bbb N$$.)