Partial order involving cartesian product: P x Q

Question

So I'm a bit confused with the concept of a partial order.

P = {1, 4}

Q = {1, 2}

How do you define a partial order Z on (P x Q)? and how would the hasse diagram of Z look like? is it even possible with these two sets?

I know a partial order has to be reflexive, antisymmetric, and transitive but I'm still confused with the idea of defining a partial order.

Thank you

Answer 1

The elements of $P\times Q$ are the ordered pairs $\langle 1,1\rangle,\langle 1,2\rangle,\langle 4,1\rangle$, and $\langle 4,2\rangle$. Make any Hasse diagram with those four entries. For instance, you could have this Hasse diagram:

          <1,1>  
            |  
          <1,2>  
            |  
          <4,1>  
            |  
          <4,2>

In that case your partial order would actually be a total (or linear) order. Or you could have this one:

          <4,2>  
          /   \  
         /     \  
      <4,1>   <1,1>  
         \     /  
          \   /  
          <1,2>

And there are many, many other possibilities. The relation corresponding to this second Hasse diagram is the set containing the following ordered pairs of elements of $P\times Q$:

$$\begin{align*}&\big\langle\langle 1,2\rangle,\langle 1,2\rangle\big\rangle,\big\langle\langle 1,2\rangle,\langle 1,1\rangle\big\rangle,\big\langle\langle 1,2\rangle,\langle 4,1\rangle\big\rangle,\big\langle\langle 1,2\rangle,\langle 4,2\rangle\big\rangle,\\ &\big\langle\langle 4,1\rangle,\langle 4,1\rangle\big\rangle,\big\langle\langle 4,1\rangle,\langle 4,2\rangle\big\rangle,\big\langle\langle 1,1\rangle,\langle 1,1\rangle\big\rangle,\big\langle\langle 1,1\rangle,\langle 4,2\rangle\big\rangle,\\ &\big\langle\langle 4,2\rangle,\langle 4,2\rangle\big\rangle \end{align*}$$

Try to see why, then try to write out the relation corresponding to the first Hasse diagram and to the one below:

    <1,1>        <1,2>        <4,1>        <4,2>