4023f10ps1

4023f10ps1 - X with 5 elements to a set Y with 7 elements?...

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Homework #1 Due: August 30 2010 1. Let A = f a; b; c; d; e g , B = f 1 ; a; b; 2 ; c g , and C = f 3 ; c; a; 1 ; f g . Answer the following questions concerning these sets. Remember that the cardinality of a set S , denoted j S j , means the number of elements in the set. Also the relative difierence of sets S and T , denoted S n T , is S n T = f x 2 S : s = 2 T g : That is, S n T consists of all of the element of S that are not elements of T . (a) Write the set A \ B . (b) Write the set A [ B . (c) Write the set A n ( B \ C ). (d) Write the set ( A \ C ) £ ( C n B ). (e) What is the cardinality of P ( A )? Remember that P ( A ) denotes the power set of A . (f) What is the cardinality of A £ C ? (g) Which set has more elements: P ( A £ C ) or P ( A ) £ P ( C )? 2. Give an example to show that if X is a subset of A £ B , then X need not be of the form C £ D where C is a subset of A and D is a subset of B . 3. How many functions are there from a set
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Unformatted text preview: X with 5 elements to a set Y with 7 elements? How many of these functions are injective? 4. (a) For every integer n 1 let f ( n ) be dened by the formula f ( n ) = P n k =1 (3 k 2). Compute f (1), f (2), and f (3). (b) Fill in the blank to make a true statement: f ( n + 1) = f ( n ) + : (c) Use induction to prove that for every integer n 1, n X k =1 (3 k 2) = n (3 n 1) 2 : 5. Let A be a 10 element subset of f 1 ; 2 ; 3 ; :::; 50 g . Show that A possesses two dierent 4-element subsets, the sums of whose elements are equal. Hint: Use the pigeon hole principle. Math 4023 1...
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This document was uploaded on 12/28/2011.

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