In my lecture slides I have two examples of sets of tuples, where the first one is in PCP while the other one is not. The first is:
$$PCP:=\{<(x_1,y_1)=(1,101), (x_2,y_2)=(10,00), (x_3,y_3)=(011,11)>\} \in PCP$$
The solution for this is $x_1 \cdot x_3 \cdot x_2 \cdot x_3=y_1 \cdot y_3 \cdot y_2 \cdot y_3$. Since the set only has $3$ elements you can use a simple trial and error method and figure out a solution. However, what if the number of tuples is larger, let's say 9 tuples? How can you show that or rather proof that no matter the arrangement of the tuples, the set is not in PCP?
PS - Since PCP is not in NP, that means that one could write a program that could check whether a given set is in PCP. However, I am looking for a more mathematical solution.
Well, the PCP is undecidable. I.e., there is no general algorithm to decide if a PCP has a solution or not.
You could use some heuristics, or simply trial and error.