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Unformatted text preview: 6 I / Preliminaries I! (1.10) EXAMPLES ,
1) If Someone rolls a die seven times, then one of the numbers from the set S = {1,2,3,4,5,6) must occur (at least) twice. if the die were rolled
only ﬁve times, then (at least) one of thenumbers in S does not occur. If __
the die were rolled six times, then each number in 5" occurs if and only if no number in 8 occurs twice. Here, we can View the elements in S as the
“pigeonholes” and the outcomes of the rolls of the die as the “pigeons.” 2) Suppose at a party (with at least two people) that no two people
shook each other’s handsmore than once. We claim that there were (at
least) two people at the party who shook the same number of hands. Let n
be the number of people at the party who shook at least one hand. If n : i},
then everyone shook the same number of hands (namely, none) and we are
done. So we may assume that n is positive and since a = 1 is impossible (it
takes two peeple to shake hands), we may assume an. 2 2. Let the pigeonholes
be the integers 1 to n — 1 and assign each of the n people who shook at least
one hand to the pigeonhole that corresponds to the number'of hands they shook. Notice that since nobody shook his own hand and no two people shook each other’s hands more than once, each of these n people shook at most 11* 1 hands. Since there are more people than pigeonholes, two people
must be assigned to the same pigeonhole', that is, two people shook the same number of hands. d _' “a”; .‘i'. .'.'.u.d.'i‘...' vi“ "stir, “‘3” ‘v .' _‘ . .m.m .' . 1w. 'i EXERCISES  J The symbol T means that a solution or a suggestion for that exercise appears ‘ é
atthe 1sack of the book. ' ¢i1.1.LetS={n€P:n2 <lOD}andietT={nEP:n23mforsome
f l ' a) Find S’ﬂT.
b) Find [8 U Tl in two ways. I t 1.2. Show that [BQI = f’P(B) by ﬁnding the elements of these two sets. T 1.3. Prcwe that SzTifand onlyifSQTandTgS. a t 1.4. Prove that s g T s and mags s u T = T. V.1.5,Provethat(AﬂC’)ﬂ(BﬂC)=AﬂBﬂC. TL6. Prove that Ah (BUG) : {AnB)U(AnG). ' 1.7. ProvethatAU(BﬂC)=(AUB)Q(AUC).  1 /Sets and Counting 7 V: t1.8. a) Prove that (A X B) n (A x C’) : A X (B00).
13) Suppose A, B, and C are ﬁnite sets. Prove that MA x B)O(A x C)! =
[A]  [B o C]. 1.9. a) Prove that A X (B U C) x (A x B) U (A x C). b) Suppose A, B, and C are ﬁnite sets. Then it follows from part (a)
that {AxrBu0)I=ItAxB)u<AxGJI. Give another proof Of this result using Proposition (1.1) and Exes.
cise 1.8{b}. 1.10. In how many different ways can one choose three elements (Without
replacement) from a set with ﬁve elements? u 1.11. Show that (f) = ( n ). TL"? 2 + I
r r m 1 r
T 1.13. If A,B, and C‘ are finite sets, prove that {AUBUCI:lAf+]Bl—t—IO—1AHBeIBﬂCl
—1An0:+[msneg. 1.12. Prove that t 1.14. Prove that among any 13 people, at least two have their birthday in
the same month. ' 1.15. Prove that if four distinct numbers are chosen from the set {1,2,3, 4,5,6}, then two of the four must have a sum of seven. (Hint: Take the pigeon—
holes to be the sets {1, 6}, {2,5}, and {3,4}.) t 1.16. Prove that if n pigeOns are assigned to m pigeonholes, then one of the
pigeonholes must contain at least — 1) + 1 pigeons, Where [st
denotes the greatest integer in a; (i.e., the integer s such that s g :1: < ' s + 1). 1.17. Suppose that S is a set consisting of nine positive integers, each of which
is lees than 50. Show that there exists (at least) two different subsets of _
S such that the sum of the integers in one subset equals the sum of the
integers in the other subset. (Hint: Use the Pigeonhole Principle with the subsets of S as the pigeons and the integers from 0 to 405 as the
pigeonholes.)  ...
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 Spring '08
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 Algebra

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