Can someone explain why this one (the top answer) is false:
For vector spaces, dim(U+V)=dimU+dimV−dim(UV), so dim(U+V+W)=dimU+dimV+dimW−dim(UV)−dim(UW)−dim(VW)+dim(UVW)
For vector spaces, dim(U+V)=dimU+dimV−dim(UV), so dim(U+V+W)=dimU+dimV+dimW−dim(UV)−dim(UW)−dim(VW)+dim(UVW)
dim(U+V) = dimU+dimV−dim(UV)
dim(U+V+W)
= dim(U+V)+dim(W) - dim((U+V)W)
= dim(U)+dim(V)+dim(W) - dim(UV) - dim(UW+VW) <-- bzzt, wrong
= dim(U)+dim(V)+dim(W) - dim(UV)-dim(UW)-dim(VW) + dim(UVW)
U = {multiples of (1,0)}
V = {multiples of (0,1)}
Oh, and A minor correction: you mean "zero" instead of "empty".
dim(U+V+W)<= dim(U)+dim(V)+dim(W) - dim(UV)-dim(UW)-dim(VW) + dim(UVW).
So the result is weakened, but not entirely lost.