By László Lovász, József Pelikán, Katalin L. Vesztergombi
Discrete arithmetic is readily turning into probably the most very important components of mathematical study, with purposes to cryptography, linear programming, coding thought and the speculation of computing. This publication is geared toward undergraduate arithmetic and desktop technological know-how scholars drawn to constructing a sense for what arithmetic is all approximately, the place arithmetic may be priceless, and what varieties of questions mathematicians paintings on. The authors talk about a few chosen effects and techniques of discrete arithmetic, ordinarily from the components of combinatorics and graph concept, with a bit quantity thought, likelihood, and combinatorial geometry. anywhere attainable, the authors use proofs and challenge fixing to aid scholars comprehend the recommendations to difficulties. furthermore, there are lots of examples, figures and workouts unfold in the course of the publication.
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Extra resources for Discrete Mathematics: Elementary and Beyond (Undergraduate Texts in Mathematics)
So the examples above suggest the following identity: n 0 2 + n 1 2 + n 2 2 + ··· + n n−1 2 + n n 2 = 2n . 4) Of course, the few experiments above do not prove that this identity always holds, so we need a proof. 52 3. Binomial Coeﬃcients and Pascal’s Triangle We will give an interpretation of both sides of the identity as the result of a counting problem; it will turn out that they count the same things, so they are equal. It is obvious what the right-hand side counts: the number of subsets of size n of a set of size 2n.
In how many ways can you send the postcards, if (a) there is a large number of each kind of postcard, and you want to send one card to each friend; (b) there is a large number of each kind of postcard, and you are willing to send one or more postcards to each friend (but no one should get two identical cards); (c) the shop has only 4 of each kind of postcard, and you want to send one card to each friend? 24 1. Let’s Count! 29 What is the number of ways to color n objects with 3 colors if every color must be used at least once?
8: n−1 n−1 n . 2). It also gives us a tool to prove many properties of the binomial coeﬃcients, as we shall see. 2. 3) using the Binomial Theorem. 2): We n can replace n0 by n−1 (both are just 1), n1 by n−1 + n−1 0 0 1 , 2 by n−1 + n−1 1 2 , etc. 6 Identities in Pascal’s Triangle + · · · + (−1)n−1 n−1 n−1 + n−1 n−2 + (−1)n 51 n−1 , n−1 which is clearly 0, since the second term in each pair of brackets cancels with the ﬁrst term in the next pair of brackets. This method gives more than just a new proof of an identity we already know.