A more graduate-level book but one I found pleasing:

A Logical Approach to Discrete Mathematics: https://www.amazon.com/Logical-Approach-Discrete-Monographs-...

And a more pragmatic approach to the same material (with a lot of cross-over in terms of proof-style, etc):

Programming in the 1990s: http://www.springer.com/gp/book/9780387973821

But one I particularly enjoyed early on was written for liberal-arts level students of maths (who might've been traumatized by maths in the past):

Introduction to Graph Theory: https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathe...

It will actually get you into writing proofs in set theory within the first couple of chapters.

I would expect no less even in an introductory text. Mathematics is meant to be precise and formal mathematics even more so. Prosaic pedagogy might suit a liberal arts introduction but it won't help the reader to convince themselves that the statements and theorems proposed are true and valid.

The syntactic approach is really useful as a programmer. A Logical Approach to Discrete Math[0] teaches programmers how to develop proofs in this style using logic and syntactic substitution. Once shown how easy it is to achieve rigour it is hoped that work-a-day programmers can formulate their own proofs and develop stronger specifications using mathematical tools.

I'm not a formally trained mathematician and am completely self-taught as a programmer but with some background in set and graph theory I found it approachable. It's certainly a good companion text to learning more practical systems like TLA+ or Lean.

[0] https://www.amazon.ca/Logical-Approach-Discrete-Math/dp/0387...

I found "A Logical Approach to Discrete Math" by Gries and Schneider to be delightful. It teaches discrete math using formal logic as the basis of the whole system.

http://www.amazon.com/Logical-Approach-Discrete-Monographs-C...