I took a brief look at Gensler's book and it looks interesting! If you're looking for something good that covers the basics in more detail, I recommend Epp's D

I took a brief look at Gensler's book and it looks interesting! If you're looking for something good that covers the basics in more detail, I recommend Epp's Discrete Mathematics with Applications. This covers material like truth tables, valid and invalid arguments, quantifiers, etc. In addition to the treatment of logic, it also has chapters on induction, combinatorics, graphs and trees, analysis of algorithms, finite state automata, etc. I chose this book when I taught a college course in "math for computer science majors", and it was a great choice overall. It's probably the most beginner-friendly of any book covering this material, and it's extremely organized and well written. Costs an arm and a leg to buy new, so buy a used copy of an older edition.