Introduction to Computer Theory 2nd Edition by Daniel I. A. Cohen, James Ed. Cohen PDF

This an excellent book. Basically, the whole point of it is to mathematically define what a computer is and prove that it works. The author does this by defining and manipulating mathematical alphabets and languages without resorting to any kind of advanced math. Starting from nothing, the whole thing leads up to Turing Machines. More specifically, according to the Preface, the goals of the book are:

1. To introduce a student of Computer Science to the need for and the working of mathematical proof...
2. To develop facility with the concepts, notations, and techniques of the theories of Automata, Formal Languages, and Turing machines...
3. To provide historical perspective on the creation of the computer with a profound understanding of some of its capabilities and limitations...

The author did a wonderful job of it. Plus, unlike almost all other computer/math books. This book is almost enjoyable to read. Again, as stated in the Preface:

"This book is written for students with no presumed background of any kind. Every mathematical concept used is introduced from scratch. Extensive examples and illustrations spell out everything in detail to avoid any possibility of confusion."