Let $n \in \mathbb N$ and the function $\text{ld}(x)$ be mapping each number with its last digit (in decimal notation). Then consider the functions $f$ and $g$ defined as follows: $$f:n\in \mathbb N \to \text{r}(\text{ld}(n), 7)\;\;\;g:n\in \mathbb N \to \text{r}(\text{ld}(n), 13)$$ knowing that $\text{r}(a,b)$ is the function that returns the remainder from the division of $a$ by $b$.
Let $\rho$ be a order relation on the set $S:=\{n\in\mathbb N \mid n < 10\}$, then $$a \;\rho\; b \Leftrightarrow a = b \text{ or } (f(a)<f(b) \text{ and } g(a)<g(b))$$
I need to
- draw the Hasse diagram relative to $(S, \rho)$;
- tell if $(S, \rho)$ is a lattice;
- find $\{4,8\}$ lower bounds and $\{3,7\}$ upper bounds.
* My attempt to soultion *
I have drawn the following diagram:
is it possible that an element has no relation to any other in the set? In that case is it correct the way I have built the diagram?
Provided that $7$ has no relations, this isn't a lattice because we can't find a $\sup$ or an $\inf$.
I think the only lower bound for $\{4,8\}$ is $0$, but there are no upper bounds for $\{3,7\}$ as $7$ isn't relatable to any element in $S$.
Is my reasoning correct? Have I done anything wrong?
Yes. A minor nitpick:
$\sup$ or $\inf$ of what? It is clear what you mean, but anyway...