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Dr. Timo de Wolff Institute of Mathematics www.math.tamu.edu/~dewolff/Fall14/math302.html MATH 302 – Discrete Mathematics – Section 501 Homework 6 Fall 2014 Due: Friday, October 24th, 2014, 9:10 a.m. When you hand in your homework, do not forget to add your name and your UIN. Exercise 1. For every r ∈ N∗ we define rZ := {r · n : n ∈ Z} 1. Give all elements of the sets 3Z, 5Z and 7Z in the interval [−20, 20]. 2. Let r ∈ N∗ fixed. Investigate the functions fr : Z → Z, x 7→ r · x gr : Z → Z, x 7→ jxk . r Prove or disprove that they injective or surjective (caution: it might depend on r). 3. Show that you can change the domain or codomain of one of the functions fr , gr such that it becomes a bijection (for all k). What can we conclude for the cardinality of rZ? Exercise 2. 1. Let S := {1, 2, 4, 7}. Compute the values of the following sums (write also down the summands): 5 X j=1 (j + 1), 4 X j=0 j (−2) , X j · (j − 1), 3 2 X X i=0 j=0 j∈S (3i + 2j) 2 X 3 X ij. i=0 j=0 P 2. Show that for every n ∈ N∗ and every a1 , . . . , an ∈ R holds: nj=1 (aj −aj−1 ) = an −a0 . Do not use a “· · · ” argument. Instead, give a direct argument via how often particular terms appear and which sign they have. Exercise 3. Prove via mathematical induction for every n ∈ N∗ : If A1 , . . . , An and B1 , . . . , Bn are sets satisfying Aj ⊆ Bj for every 1 ≤ j ≤ n, then n \ Aj ⊆ j=1 n \ Bj . j=1 Exercise 4. Prove via mathematical induction for every n ∈ N: If S is a finite set with cardinality n, then the power set P(S) has cardinality 2n . 1