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Dr. Timo de Wolff Institute of Mathematics www.math.tamu.edu/~dewolff/Spring16/math302.html MATH 302 – Discrete Mathematics – Section 501 Homework 6 Spring 2016 Due: Wednesday, March 23rd, 2016, 4:10 pm. 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 {..., −4, −2, 0, 2, 4, ...}. : n ∈ Z}. E.g. 2Z = 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 the functions injective or surjective. Caution: it might depend on the choice of r; distinct different cases if necessary.. 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 r). Use this fact to conclude 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): 4 X j=0 j (−2) , X j · (j − 1), 2 X 3 X i=0 j=0 j∈S (3i + 2j) 3 2 X X ij. i=0 j=0 P 2. Show that for every n ∈ N∗ and every a1 , . . . , an ∈ R it holds that nj=1(aj − aj−1 ) = an − a0 . Do not use an argument involving “· · · ”. Instead, give a direct, prose argument exploiting 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