A decent textbook for discrete mathematics that I would recommend is 'Discrete Mathematics with Applications' by Susanna S. Epp. It contains a lot of examples and is quite easy to follow.
What is a good book for the subject of discrete mathematics?
This is kind of vague. Do you mean "discrete mathematics" in the sense the undergrad often entails? A very brief survey of set theory, combinatorics, graph theory, etc? If so, try Rosen. If not, be more spefic.