By László Lovász, József Pelikán, Katalin L. Vesztergombi
Discrete arithmetic is readily turning into some of the most vital parts of mathematical examine, with purposes to cryptography, linear programming, coding concept and the speculation of computing. This ebook is aimed toward undergraduate arithmetic and desktop technological know-how scholars drawn to constructing a sense for what arithmetic is all approximately, the place arithmetic might be invaluable, and what varieties of questions mathematicians paintings on. The authors speak about a couple of chosen effects and strategies of discrete arithmetic, commonly from the components of combinatorics and graph conception, with a bit quantity concept, chance, and combinatorial geometry. anywhere attainable, the authors use proofs and challenge fixing to aid scholars comprehend the suggestions to difficulties. moreover, there are many examples, figures and workouts unfold during the booklet.
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Extra info for Discrete Mathematics: Elementary and Beyond (Undergraduate Texts in Mathematics)
The last box is labeled “New Yorkers having 500,000 strands of hair”. Now if everybody goes to the proper box, then about 10 million New Yorkers are properly assigned to some box (or hole). Since we have only 500,001 boxes, there certainly will be a box containing more than one New Yorker. This statement is obvious, but it is very often a powerful tool, so we formulate it in full generality: If we have n boxes and we place more than n objects into them, then there will be at least one box that contains more than one object.
2)) that this proves the assertion about the sum of the ﬁrst n odd numbers. In other words, what we have actually shown is that if the assertion is true for a certain value (n − 1), then it is also true for the next value (n). This is enough to conclude that the assertion is true for every n. We have seen that it is true for n = 1; hence by the above, it is also true for n = 2 (we have seen this anyway by direct computation, but this shows that this was not even necessary: It follows from the case n = 1).
N − k + 1) n−1 n−2 n−k+1 These factors are quite simple, but it is still diﬃcult to see how large their product is. The individual factors are larger than 1, but (at least at the beginning) quite close to 1. But there are many of them, and their product may be large. 3 We get ln nk n(n − 1) · · · (n − k + 1) = ln + ln n n + ln n−1 n−2 n . 71828 . . ) This way we can deal with addition instead of multiplication, which is nice; but the terms we have to add up became much uglier! What do we know about these logarithms?