From these two steps, mathematical induction is the rule from which we infer that the given statement is established for all natural numbers. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs.
The simplest and most common form of mathematical induction infers that a statement involving a natural number n holds for all values of n. The proof consists of two steps: The basis (base case): prove that the statement holds for the first natural number n. Usually, n = 0 or n = 1.The inductive step: prove that, if the statement holds for some natural number n, then the statement holds for n + 1.
This book is an introduction to combinatorial mathematics, also known as combinatorics. The book focuses especially, but not exclusively, on the part of combinatorics that mathematicians refer to as \"counting.\" The book consists almost entirely of problems. Some of the problems are designed to lead you to think about a concept, others are designed to help you figure out a concept and state a theorem about it, while still others ask you to prove the theorem. Other problems give you a chance to use a theorem you have proved. From time to time there is a discussion that pulls together some of the things you have learned or introduces a new idea for you to work with. Many of the problems are designed to build up your intuition for how combinatorial mathematics works. Above all, this book is dedicated to the principle that doing mathematics is fun. As long as you know that some of the problems are going to require more than one attempt before you hit on the main idea, you can relax and enjoy your successes, knowing that as you work more and more problems and share more and more ideas, problems that seemed intractable at first become a source of satisfaction later on.
There are six chapters as well as an appendix with three additional topics: What is CombinatoricsApplications of Induction and Recursion in Combinatorics and Graph TheoryDistribution ProblemsGenerating FunctionsThe Principle of Inclusion and ExclusionGroups Acting on SetsThe three supplemental sections deal with relations, mathematical induction, and exponential generating functions. 1e1e36bf2d