We have a set S= {1,2,3,4} with following relation: aRb <-> a ≤ b ≤ a^2
With focus on the relation we get following partially ordered set:
{1,2},{1,3},{1,4},
{2,3},{2,4},
{3,4}
which gives the following hasse diagram.
4
|
3
|
2
|
1
also we get a straight line from 1 to 4.
But i am a bit unsure if this is right, so am i thinking right?
Your list of ordered pairs isn’t right, I’m afraid. First, you’ve omitted the pairs $\langle 1,1\rangle,\langle 2,2\rangle,\langle 3,3\rangle$, and $\langle 4,4\rangle$, all of which are in $R$: for any positive integer $n$ it’s true that $n\le n\le n^2$. Next, you have a lot of pairs that aren’t in $R$. For instance, it’s not true that $1\le 2\le 1^2$, so $\langle 1,2\rangle\notin R$. Similarly, $\langle 1,3\rangle\notin R$ and $\langle 1,4\rangle\notin R$. It is true, though, that $2\le 3\le 2^2$ and $2\le 4\le 2^2$, so you’re correct in thinking that $\langle 2,3\rangle$ and $\langle 2,4\rangle$ are in $R$. Similarly, $3\le 4\le 3^2$, so $\langle 3,4\rangle\in R$, as you said.
This is going to change your Hasse diagram a bit; I’ll let you see if you can correct on your own, but leave a question if you have problems.