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 .
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