145HW2

# 145HW2 - B k and use the inclusion-exclusion formula 5 100...

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Homework assignment Number 2 Math 145 // due on 01/19/2007 1. Problems Section 2.5: 1, 5, 7, 8. 2. n persons can arbitrarily shake hands with one another. Show that there is always a pair of persons that shake the same number of hands. 3. Consider a set of 6 points on which we draw a complete graph, that is we draw an edge between any two points. We then color each edge arbitrarily red or blue. Show that you can ﬁnd a triangle whose edges are all the same color. 4. Consider two sets A and B such that ±A = n and ±B = k. A map f : A ±→ B is onto iﬀ b B, a A, f ( a ) = b. We also say that f is a surjection. Compute the number of surjections f : A ±→ B . Hint: assume that B = { 1 , . . . , k } . Set F k = { f : A ±→ B, f ( A )
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Unformatted text preview: B \ { k }} and use the inclusion-exclusion formula. 5. 100 people are at a party. Each has an even number of aquaintances (possibly 0). Show that there are at least three persons having the same number of acquaintances in the group. 6. Let A = { 1 , . . . , n } . Compute the number of maps f : A ±→ A without a ﬁxed point ( f ( i ) ² = i, ∀ i = 1 , . . . , n ) . 7. Let a 1 , a 2 , . . . , a n be n given integers not necessarily distinct. Show that there exist integers k, l with 0 ≤ k < l ≤ n such that a k +1 + a k +2 + a l is a multiple of n. Hint: consider the sums S i = ∑ i j =1 a j , i ≤ n . 1...
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## This homework help was uploaded on 04/08/2008 for the course MATH 145 taught by Professor Peche during the Spring '07 term at UC Davis.

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