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Lecture Notes on Discrete Mathematics

The abuse of technology is evident in some of the examples cited above. Fingers-on-the-buttons has now replaced engagement-of-the-brain with in- creasing frequency. I should also call attention to the insistence in educa- tional documents that students in K-4 be allowed to have access at all times to the calculator ([MAF], pp.57-59; [N1], p.19; [N5], p.vi; [UN]). In theory, giving children an extra tool in the form of a calculator can do nothing but “empower” them mathematically, to use a term that is popular these days, but what if the theory is not born out by hard facts? In anecdote after anec- dote, one hears horror stories of the. | Lecture Notes on Discrete Mathematics A. K. Lal September 26 2012 2 Contents 1 Preliminaries 5 1.1 Basic Set Theory. 5 1.2 Properties of Integers . 8 1.3 Relations and Partitions. 16 1.4 Functions. 21 2 Counting and Permutations 27 2.1 Principles of Basic Counting. 27 2.1.1 Distinguishable Balls. 27 2.1.2 Indistinguishable Balls and Distinguishable Boxes . 36 2.1.3 Indistinguishable Balls and Indistinguishable Boxes. 39 2.1.4 Round Table Configurations. 40 2.2 Lattice Paths. 41 2.2.1 Catalan Numbers . 43 2.3 Some Generalizations. 46 2.3.1 Miscellaneous Exercises . 50 3 Advanced Counting 53 3.1 Pigeonhole Principle. 53 3.2 Principle of Inclusion and Exclusion. 58 4 Polya Theory 63 4.1 Groups . 63 4.2 Lagrange s Theorem. 73 4.3 Group Action . 78 4.4 The Cycle Index Polynomial . 82 4.4.1 Applications . 84 4.4.2 Polya s Inventory Polynomial. 86