(Question too long to post as title) The question is, Let S = {1,2,3,4,5,6,7,8,9} and let T = {2,4,6,8}. Let R be the relation on P(S) defined by for all X, Y ∈ P (S), (X, Y ) ∈ R if and only if |X − T| = |Y − T| and i have to find the number of equivalence classes. I had to initially prove it was an equivalence relation which i dont think i had trouble with but i'm struggling to understand how to approach solving this? I looked up what an equivalence class is again and it makes sense when it was such a simple example like for instance when a set of the relation was given to me and all i had to do was check for each x in the original set which other x did it have a relation with and then find the distinct relations. However, i am struggling to understand how to apply it here to such a larger scale and more complex problem.
In this case two elements or subsets of S in P(S) are related if the elements in X and not T give a cardinality equal to the elements in Y and not T? is this the correct understanding? Then to find the number of equivalence classes i have to do what? I dont really get how to start the process any help is appreciated thank you!
Could someone perhaps explain what an equivalence class would mean in this case?
Hint: for any subset $X, |X-T|$ is the number of odd elements of $X$. How many choices are there among the subsets of $S$ for the number of odd elements?