Hi there, I'm struggling a bit with a uni question I've been given. I think I understand the basics but possibly not!! Any help greatly appreciated!
Take a standard 8×8 chessboard, and label the squares
(i, j) 1 ≤ i ≤ 8, 1 ≤ j ≤ 8, with (1, 1) being the bottom lefthand square,
and (8, 8) the top righthand square. Define a relation on these squares by
(i, j) ∼ (i′, j′) iff i + j = i′ + j′.
(a) Show that ∼ defines an equivalence relation on the chessboard.
(b) How many equivalence classes are there, and what are their sizes?
(c) Choose equivalence class representatives for the classes.
Here are my thoughts...
a) would I assume ~ if i+j not equal to i'+j' and show this is not the case?
b) I'm not entirely sure but I would assume that as the canonical projection map is surjective and that there are 64 ways to write i+j that there are 64 equivalence classes of size 2, then again I'm well prepared to believe I've missed the point!
c) As a set of class representatives is a subset of X which contains exactly one element from each equivalence class could the class representative just be either the i's or the j's
Luckily this is over the computer so if people tell me I'm completly wrong you won't see me blushing!!
Thanks for the help, Laura