Problem 2

Does there exist a finite set

M

of points in space, not all in one plane, such that for every two points

A,B\in M

there are two other points

C,D\in M

for which the lines

AB

and

CD

are parallel but distinct?

Step 5 of 5: Conclusion

In plain words

The construction is finite and its two planes witness non-coplanarity.

M\text{ is finite and non-coplanar}

Detailed analysis

The union has finitely many points and contains points in two perpendicular planes, so it is not coplanar. The answer is yes.