In my university, they used "Discrete Mathematics and Its Application" by Kenneth Rosen. If I remember correctly, the main book has solutions for all odd-numbered (or even-numbered?) exercises, and there is a special solutions volume.

I would add that sets and relations are so ubiquitous in mathematics, that if you just take regular courses, you will definitely become better at them. A book specifically in discrete math is needed for topics like graphs, counting techniques (such as inclusion-exclusion principle and combinatorics), Boolean logic, etc.