Partial orders - Question

Good day to all,

We have just started relations and I came across the following question in my textbook:

**Let X = {1, 2}. List all the partial orders that can be defined on X.**

My solution:

I began by computing X x X (Cartesian product) which gave me:

X x X ={(1,1), (1,2), (2,1), (2,2)}

Since we are looking for partial orders, the relation has to be simultaneously reflexive, antisymmetric and transitive.

If the relation R on X is reflexive then it must contain (1,1) and (2,2)

The ordered pairs (1,2) and (2,1) cannot belong to R for if they did and R is also antisymmetric then that would imply 1=2, which is false.

Therefore I concluded that the list of partial orders is the set:

{(1,1), (2,2)}

I was wondering if my logic is flawed and if so what errors have I committed?

Finally, is it possible in this problem to determine the number of partial orders (cardinality)?

Any advice would be greatly appreciated.

Kindest regards