By L. Lovasz, J. Pelikan, K. Vesztergombi

**Read or Download Discrete Mathematics. Elementary and Beyond PDF**

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**Additional info for Discrete Mathematics. Elementary and Beyond**

**Example text**

Prove that A ∪ (B ∩ C) = (A ∪ B) ∩ C. Show by an example that the condition that A is a subset of C cannot be omitted. 18 What is the diﬀerence A \ B if (a) A is the set of primes and B is the set of odd integers? (b) A is the set of nonnegative real numbers and B is the set of nonpositive real numbers? 19 Prove that for any three sets A, B, C, ((A \ B) ∪ (B \ A)) ∩ C = ((A ∩ C) ∪ (B ∩ C)) \ (A ∩ B ∩ C). 8 The Number of Subsets of a Given Size 23 denote the set of all 2-element subsets of A. 20 Let A be a set and let A 2 Which of the following statements is true?

Finally there are 3 boys who like all four activities. In addition we know that everybody likes at least one of these activities. How many boys are there in the class? 4 Pigeonholes Can we ﬁnd in New York two persons having the same number of strands of hair? One would think that it is impossible to answer this question, since one does not even know how many strands of hair there are on one’s own head, let alone about the number of strands of hair on every person living in New York (whose exact number is in itself quite diﬃcult to determine).

Nk positions for letter Z. Having formulated it this way, we can see that this is nothing but the question of distributing n presents to k children when it is prescribed how many presents each child gets. Thus we know from the previous section that the answer is n! n2 ! · · · nk ! ) = 180). We say that these words are “essentially the same” (at least as far as counting anagrams goes): They have two letters repeated twice and two letters occurring only once. ” (a) How many 6-letter words are there, if, to begin with, we consider any two words diﬀerent if they are not completely identical?