consider the setting depicted in this last image. The 3 x 3 matrix
has been altered and only seven positions remain. This version
involves an odd number of moves, but it is not obvious that there
is a map between the 3 x 3 and this version that constitutes
these examples have sharpened your intuitions concerning the
ideas of isomorphism, homomorphism and functional equivalence.
If it hasn't already occurred to you, let me point out that determining
whether or not any of these relations hold between two systems
is usually not a simple matter.
if the systems under study have not already been appropriately
characterized mathematically, we can't even ask the question.
And, even then answering the question is nontrivial.