Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here. Table of contents : Cover Page 1 Title Page Page 5 Copyright Page 6 Contents Page 9 To the Student Page 19 To the Instructor Page 25 Acknowledgments Page 27 1 Joy Page 31 2 Speaking and Writing of Mathematics Page 32 3 Definition Page 34 4 Theorem Page 38 5 Proof Page 45 6 Counterexample Page 53 7 Boolean Algebra Page 55 Chapter 1 Self Test Page 60 8 Lists Page 63 9 Factorial Page 70 10 Sets I: Introduction, Subsets Page 73 11 Quantifiers Page 81 12 Sets II: Operations Page 86 13 Combinatorial Proof: Two Examples Page 96 Chapter 2 Self Test Page 14 Relations Page 15 Equivalence Relations Page 16 Partitions Page 17 Binomial Coefficients Page 18 Counting Multisets Page 19 Inclusion-Exclusion Page Chapter 3 Self Test Page 20 Contradiction Page 21 Smallest Counterexample Page 22 Induction Page 23 Recurrence Relations Page Chapter 4 Self Test Page 24 Functions Page 25 The Pigeonhole Principle Page 26 Composition Page 27 Permutations Page 28 Symmetry Page 29 Assorted Notation Page Chapter 5 Self Test Page 30 Sample Space Page 31 Events Page 32 Conditional Probability and Independence The core consists of Sections 1 through 24 optionally omitting Sections 18 and I hope you find this supplement helpful.
Please send me your feedback by email to ers jhu. Thank you. This section may be assigned as reading. The purpose of this section is to instill in students a sense of the pleasure mathematical work can bring. I recommend that you assign the one problem contained herein but not for course credit. Admonish your students thoroughly not to discuss the problem with each other or else they will spoil the experience.
Do not give them hints. Be encouraging and reassure them that when they have found the answer, they will know they are correct. This section may also be assigned as reading. The goal here is to emphasize that clarity of language is vital in mathematics. The extent to which students can articulate their thoughts is a good indicator of how well they understand them.
Furthermore, the very act of putting their thoughts into clear sentences helps with the learning process. They also believe that mathematics instructors will accept lousy writing, horrible penmanship, and innumerable crossouts on crumpled paper that has been torn out of a spiral notebook. We need to expect a lot better! Mathematics: A Discrete Introduction Here are written instructions. You will find before you six pieces: a large isosceles right triangle, a small isosceles right triangle, a square, a parallelogram, a trapezoid with one side perpendicular to the parallel sides , and an oddly shaped nonconvex pentagon.
Place the square in the upper right with its sides vertical and horizontal. Place the trapezoid to the left of the square so that its long parallel side aligns with the top of the square and its short parallel side aligns with the bottom of the square. Note that the side perpendicular to the parallel sides exactly abuts the left side of the square. Place the parallelogram so that one of the long sides of the parallelogram matches the long non-parallel side of the trapezoid.
Thus, the parallelogram is just below the left portion of the trapezoid. On the left, the short side of the parallelogram should complete a right angle with the long parallel side of the trapezoid.
Place the small right triangle below the square. One of the legs of the small right triangle exactly aligns with the bottom of the square and the other leg of the right triangle faces the inside of the puzzle.
The hypotenuse of the small right triangle should slope from the lower left to the upper right. Hold the large right triangle so that its legs are vertical and horizontal, and its right angle is to the lower right. Half of the hypotenuse of the large right triangle should align with all of the hypotenuse of the small right triangle. Finally, hold the pentagon so that its right angle is in the lower left and the two sides that form that right angle are vertical and horizontal.
It will now slide so that its right edge matches the lower half of the hypotenuse of the big right triangle and its upper edge meets up with the lower, long side of the parallelogram.
If we think of mathematics as a dramatic production, there are three main characters: Definition, Theorem, and Proof.
Important, supporting roles in this show are Conjecture and Example. Thus the first few sections of this book are dedicated to introducing these main characters. Perhaps conspicuous by its absence is Axiom. The term is introduced later in the book. This omission is intentional. In my philosophy of mathematics, axioms are a form of definition. For example, the axioms of Euclidean geometry form the definition of the Euclidean plane.
Likewise the so-called axioms of group theory are actually the definition of a group. And while we are speaking of philosophy, I try to be rather clear about mine in the text. To me, mathematics is a purely mental construction. Definitions, theorems, proofs, etc. If you agree with me, great. If you disagree with me e. You can use the text in a point-counterpoint discussion. Ultimately, I do not think these philosophical questions have much bearing on the day-to-day work of mathematicians.
In any case, the essential point to convey from this section is that mathematical definitions must be much clearer than ordinary definitions. Try having students define chair or love; as mathematicians we are lucky to be able to define our terms precisely. Furthermore, we cannot in this book define everything down to first principles. It is much too difficult for students on this level to go reduce everything to axiomatic set theory.
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