Dr. Timo de Wolff
Institute of Mathematics
www.math.tamu.edu/~dewolff/Spring16/math302.html
MATH 302 – Discrete Mathematics – Section 501
Homework 5
Spring 2016
Due: Wednesday, March 9th, 2016, 4:10 pm.
When you hand in your homework, do not forget to add your name and your UIN.
Exercise 1. Set operations then have a long runtime in computers. Thus, storing sets
without an ordering in a computer is not very effective. An alternative way is to represent
finite sets is to equip the universal (countable) set U = {a1 , . . . , ak } with an arbitrary total
ordering a1 < a2 < . . . < ak and represent the particular set by a k-tuple with entries 0
and 1. We call such tuples 01-vectors. More precisely, the j-th entry of a such a tuple is 1
if and only if the j-th element of the universe, aj , is contained in the particular set, which
is represented by the tuple. The j-th entry is 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 well as A ∩ B, A ∪ B
and A \ B as k-tuples with entries 0 and 1.
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 the 01-vector representation
discussed above. That means, you need to translate two times in your program.
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 the graph defined
here coincides the usual notion of the graph of a function that you know e.g. from
your calculus lectures.
3. Show: if f : A → B and g : B → C and g ◦ f is their composition, then
G(g ◦ f ) = {(a, c) : ∃ b ∈ B : (a, b) ∈ G(f ) and (b, c) ∈ G(g)}.
1
Exercise 3. The floor function ⌊·⌋ : R → Z assigns to every real x the largest integer k,
which satisfies k ≤ x.
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. Do not use a short operation like ⌊·⌋ :=
x − (x%1); use only elemental methods.
3. Show that the floor function is surjective but not injective.
Hint: Essentially, 1. solves 2. and vice versa. Make use of that!
2