SOLUTIONS FOR THE EXERCISES AT THE END OF THE SECTIONS AND THE REVIEW EXERCISES Warning The following solutions for the exercises listed in this book should be used only as a guide. If you have not read the section "To the Reader" in the Preface, here it is once more! These solutions should only be used as a guide. Indeed, learning to construct proofs is like learning to play tennis. It is useful to have someone teaching us the basics and it is useful to look at someone play- ing, but we need to get into the court and play if we really want to learn. Therefore, we suggest that you, the reader, set aside a minimum time limit for yourself to construct a proof without looking at the solution (as a starting point, you could give yourself one hour, and then adjust this limit to fit your ability). If you do not succeed, read only the first few lines of the proof pre- sented here, and then try again to complete the proof on your own. If you are not able to do so, read a few more lines and try once more. If you need to read the whole proof, make sure that you understand it, and after a few days try the exercise again on your own. The discussion part of a proof is enclosed in parentheses. SOLUTIONS FOR THE EXERCISES AT THE END OF THE SECTIONS Chapter 2: Basic Techniques to Prove If/Then Statements What Does "If/Then" Mean?