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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 5 Fall 2014 Due: Friday, October 17th, 2014, 9:10 a.m. When you hand in your homework, do not forget to add your name and your UIN. Exercise 1. Storing sets in a computer unordertly is not very effective since set operations then have a long runtime. An alternative way is to represent finite sets is to equip the universal (countable) set with a total ordering and represent the particular set by 01-vectors. I.e., the j-th entry of a such a vector is 1 if and only if the j-th element of the universe is contained in the particular set, which is represented, and 0 otherwise. In practise, one represents sets A1 , . . . , Ar by k-tuples, where k is the maximal element in A1 ∪ · · · ∪ Ar . 1. Let A := {2, 4, 5, 7} and B := {1, 2, 3, 5}. Represent both sets as 01-vector as well as A ∩ B, A ∪ B and A \ B. 2. Let k ∈ N be fixed. Write a pseudocode that takes two sets A, B ⊆ {0, . . . , k} as input and returns A ∩ B such that the computation is done via 01-vector representation. Exercise 2. For a function f : A → B the graph G(f ) of f is the set {(a, b) : a ∈ A and f (a) = b} ⊆ A × B. 1. Give the graphs for f : {0, . . . , 5} → Q, x 7→ x3 and g : {0, . . . , 5} → N, n 7→ fibn , where fibn denotes the n-th Fibonacci number. 2. Draw the graph of the function f : R → R, x 7→ x3 . Explain, why it coincides the usual notion from calculus and earlier. 3. Show: if f : A → B and g : B → C, then G(g ◦ f ) = {(a, c) : ∃ b ∈ B : (a, b) ∈ G(f ) and (b, c) ∈ G(g)}. Exercise 3. The floor function ⌊·⌋ : R → Z assigns to every real x the largest integer k, which satisfies k ≤ x. 1 1. Show that for every x ∈ R there exists a maximal k ∈ Z such that 0 ≤ x − k < 1. I.e., ⌊·⌋ is well defined. 2. Write a pseudocode for the floor function. 3. Show that the floor function is surjective but not injective. Hint: Essentially, 1. solves 2. and vice versa. Make use of that! 2