Monday June 2

To any partition P of X, we associated the natural surjection X ->> P.
To any function f : X -> Y, we associated a partition of X, where the cells are the preimages of elements of image(f).

This lets us refine our previous theorem about factoring functions f : X -> Y;
any function factors uniquely as X -natural surjection->> P -> S -inclusion-> Y,
where P is a partition of X, S is a subset of Y, and P->S is 1:1 and onto;
in particular, P must be the partition associated to f, and S must be the image of f.

We defined relations, and associated to any partition of X, a relation ~_P from X to X.
Then pointed out three properties it has: reflexive, symmetric, transitive.
Any relation with those properties is an equivalence relation.
Then we looked at a bunch of relations, and figured out how they failed to be equivalence relations.

Q. Is the following an equivalence relation on N?
a ~ b iff there exists an m such that a | b^m, b | a^m.