By László Lovász, József Pelikán, Katalin L. Vesztergombi
Discrete arithmetic is instantly turning into some of the most very important parts of mathematical examine, with functions to cryptography, linear programming, coding concept and the idea of computing. This ebook is geared toward undergraduate arithmetic and laptop technological know-how scholars drawn to constructing a sense for what arithmetic is all approximately, the place arithmetic may be useful, and what types of questions mathematicians paintings on. The authors speak about a couple of chosen effects and strategies of discrete arithmetic, normally from the parts of combinatorics and graph idea, with a bit quantity concept, likelihood, and combinatorial geometry. at any place attainable, the authors use proofs and challenge fixing to assist scholars comprehend the strategies to difficulties. furthermore, there are many examples, figures and routines unfold in the course of the ebook.
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Additional resources for Discrete Mathematics: Elementary and Beyond (Undergraduate Texts in Mathematics)
The number of k-subsets of an n-set is such an important quantity that there is a special notation for it: nk (read “n choose k”). Thus n k = n! (n − k)! 6) The number of diﬀerent lottery tickets is 90 5 , the number of handshakes 7 at the start of Alice’s birthday party is 2 , etc. 1 we will see why). The value of nn is 1, since an n-element set has exactly one n-element subset, namely itself. It may look a bit more tricky to ﬁnd that n0 = 1, but it is just as easy to explain: Every set has a single 0-element subset, namely the empty set.
Nn−1 , since (ignoring the factor 1 again) n! is the product of n − 1 factors, each of which is at most n. 3) 2n−1 ≤ n! ≤ nn−1 . These bounds are very far apart; for n = 10, the lower bound is 29 = 512, while the upper bound is 109 (one billion). 3). Which is larger, n! or 2n ? In other words, does a set with n elements have more permutations or more subsets? For small values of n, subsets are winning: 21 = 2 > 1! = 1, 22 = 4 > 2! = 2, 23 = 8 > 3! = 6. But then the picture changes: 24 = 16 < 4!
If you can’t ﬁnd the answer, read the following section and return to this exercise afterwards. 48 3. 2. How to distribute n pennies to k children? 4 Distributing Money Instead of distributing presents, let’s distribute money. Let us formulate the question in general: We have n pennies that we want to distribute among k kids. Each child must get at least one penny (and, of course, an integer number of pennies). How many ways can we distribute the money? Before answering this question, we must clarify the diﬀerence between distributing money and distributing presents.
Discrete Mathematics: Elementary and Beyond (Undergraduate Texts in Mathematics) by László Lovász, József Pelikán, Katalin L. Vesztergombi