By Lovasz L., Vesztergombi K. L., Pelikan J.
This publication is geared toward undergraduate arithmetic and machine technology scholars attracted to constructing a sense for what arithmetic is all approximately, the place arithmetic will be invaluable, and what types of questions mathematicians paintings on. The authors speak about a few chosen effects and techniques of discrete arithmetic, in most cases from the components of combinatorics and graph concept, with a bit quantity conception, chance, and combinatorial geometry. at any place attainable, the authors use proofs and challenge fixing to aid scholars comprehend the ideas to difficulties. additionally, there are various examples, figures, and workouts unfold during the booklet.
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Additional resources for Discrete Mathematics: Elementary and Beyond
9 Prove that the sum of the ﬁrst n powers of 2 (starting with 1 = 20 ) is 2n − 1. ” instead of further repetition. There is nothing wrong with this, if the argument is suﬃciently simple so that we can intuitively see where the repetition leads. ” in cases where the outcome of the repetition is not so transparent. The precise way of doing this is using induction, as we are going to illustrate by revisiting some of our results. 1 (recall that the answer is 2n ). As the Principle of Induction tells us, we have to check that the assertion is true for n = 0.
120. It is easy to see that as n increases, n! grows much faster than 2n : If we go from n to n + 1, then 2n grows by a factor of 2, while n! grows by a factor of n + 1. 2 Use induction to make the previous argument precise, and prove that n! > 2n if n ≥ 4. There is a formula that gives a very good approximation of n!. We state it without proof, since the proof (although not terribly diﬃcult) needs calculus. 1 [Stirling’s formula] n n√ 2πn. 14 . . 718 . . is the base of the natural logarithm, and ∼ means approximate equality in the precise sense that n!
Question: How many students in the class have no picture of any of the rock groups? First, we may try to argue like this: There are 40 students altogether in the class; take away from this the number of those having Beatles pictures (18), those having Rolling Stones picture (16), and those having Elvis pictures (12); so we take away 18 + 16 + 12. We get −6; this negative number warns us that there must be some error in our calculation; but what was not correct? We made a mistake when we subtracted the number of those students who collected the pictures of two groups twice!