# An Invitation to Discrete Mathematics by Jiri Matousek, Jaroslav Nesetril

By Jiri Matousek, Jaroslav Nesetril

This e-book is a transparent and self-contained advent to discrete arithmetic. Aimed regularly at undergraduate and early graduate scholars of arithmetic and machine technological know-how, it truly is written with the objective of stimulating curiosity in arithmetic and an energetic, problem-solving method of the provided fabric. The reader is ended in an knowing of the fundamental ideas and strategies of truly doing arithmetic (and having enjoyable at that). Being extra narrowly targeted than many discrete arithmetic textbooks and treating chosen subject matters in an strange intensity and from numerous issues of view, the booklet displays the conviction of the authors, lively and across the world popular mathematicians, that crucial achieve from learning arithmetic is the cultivation of transparent and logical pondering and conduct priceless for attacking new difficulties. greater than four hundred enclosed routines with a variety of trouble, lots of them observed via tricks for resolution, aid this method of instructing. The readers will savour the full of life and casual kind of the textual content followed via greater than 2 hundred drawings and diagrams. experts in quite a few components of technology with a uncomplicated mathematical schooling wishing to use discrete arithmetic of their box can use the ebook as an invaluable resource, or even specialists in combinatorics might sometimes study from tips to learn literature or from displays of contemporary effects. Invitation to Discrete arithmetic should still make a pleasant interpreting either for newbies and for mathematical professionals.

the most themes contain: straightforward counting difficulties, asymptotic estimates, in part ordered units, uncomplicated graph concept and graph algorithms, finite projective planes, straight forward likelihood and the probabilistic technique, producing services, Ramsey's theorem, and combinatorial purposes of linear algebra. common mathematical notions going past the high-school point are completely defined within the introductory bankruptcy. An appendix summarizes the undergraduate algebra wanted in many of the extra complex sections of the ebook.

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In words, to ﬁnd the value of h(x), we ﬁrst apply f to x and then we apply g to the result. The function h (check that h is indeed a function) is called the composition of the functions g and f and it is denoted by g ◦ f . We thus have g ◦ f (x) = g f (x) for each x ∈ X. The composition of functions is associative but not commutative. For example, if g ◦ f is well deﬁned, f ◦ g need not be. In order that two functions can be composed, the “middle set” must be the same. Composing functions can get quite exciting.

In this problem, you can test your ability to discover simple but “hidden” solutions. Divide the following ﬁgure into 7 parts, all of them congruent (they only diﬀer by translation, rotation, and possibly by a mirror reﬂection). All the bounding segments in the ﬁgure have length 1, and the angles are 90, 120, and 150 degrees. 3 Mathematical induction and other proofs Let us imagine that we want to calculate, say, the sum 1 + 2 + 22 + n i 23 + · · · + 2n = i=0 2 (and that we can’t remember a formula for the sum of a geometric progression).

What is it? 4 Functions 25 12. Let n1 , n2 , . . , nk be natural numbers, each of them at least 1, and let n1 +n2 +· · ·+nk = n. Prove that n21 +n22 +· · ·+n2k ≤ (n−k+1)2 +k−1. k “Solution”: In order to make i=1 n2i as large as possible, we must set all the ni but one to 1. The remaining one is therefore n − k + 1, and in this case the sum of squares is (n − k + 1)2 + k − 1. Why isn’t this a valid proof? ∗ Give a correct proof. 13. 2. 14. ∗ Let n > 1 and k be given natural numbers. Let I1 , I2 , .