Describing orders on sets $\{1,2,3,4\}$ and $\mathbb N$ with minimal/maximal/maximum/minimum elements

Question

1. Describe all the partial orders on $\{1,2,3,4\}$ where the set of minimal elements are $\{2,4\}$ and the set of maximal elements is $\{1,3\}$

2. Describe all the partial orders on $\{1,2,3,4\}$ where 2 is the minimum and 1 is the maximum.

3. Describe the total orders over $\mathbb N$ with a maximum and no minimum.

4. Describe the total orders over $\mathbb N$ with no maximum and no minimum.

No need to prove and we can describe it with a graph.

I'm not sure I understand the question, well, for 1. I think the set is $A=\{\text{x is even}: x> 2 , x \in \mathbb Q \}\cup \{\text{x is odd}: x<3, x \in \mathbb Q\}$

For 2, I think it's $\{1\}$ on it's own and $\{x \in \mathbb Q: x\le 2 $}

3: it's like $\omega+0$ but there should be no minimum so I'm not sure how to cut it off. Probably 4 has the same principle.

Answer 1

The question asks you to describe all the partial orders with certain properties.

For example, all the partial orders of the set $\{1,2,3\}$ whose minimum element is $1$ are:

2   3     2    3
 \ /      |    |
  1       3    2
          |    |
          1    1

The first two parts of the question give similar description of orders of $\{1,2,3,4\}$.

The last two are more difficult, since there are continuum many linear orders of $\Bbb N$, with or without a maximum. But recall that every linear order can be realized as a subset of $\Bbb Q$, so by removing a particular set we can get linear orders.