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