Dr. Timo de Wolff
Institute of Mathematics
www.math.tamu.edu/~dewolff/Spring16/math302.html
MATH 302 – Discrete Mathematics – Section 501
Midterm 2 Preparation Homework
Spring 2016
Exercise 1. Compute the following sets:
1. ({3, 6, 4, 1} \ {2, 3, 6}) ∪ ({8, 1, 3} ∩ {1, 2.5, 8})
2. ({x ∈ N : x is prime } ∩ {x ∈ Z : x is even }) ∪ {7, 9, 10}
3. {x ∈ N : ∃z ∈ N : x = 3z} ∩ {x ∈ N : ∃z ∈ N : x = 2z} ∩ {x ∈ N : ∃z ∈ N : x = 5z}
Exercise 2. Prove the de Morgan law Ac ∩ B c = (A ∪ B)c .
Exercise 3. Let
1
Z := {..., −2, −1.5, −1, −0.5, 0, 0.5, 1, 1.5, 2, ...} =
2
3 2 1 0 1 2 3
..., − , − , − , , , , , ... .
2 2 2 2 2 2 2
denote the set of half integers. Recall that an injective function is also called 1-to-1.
More generally, for k ∈ N a function f : S → T is called k-to-1 if for every t ∈ T there
exist exactly k elements s1 , . . . , sk such that f (s1 ) = f (s2 ) = · · · = f (sk ) = t.
Show that the function
1
f : Z \ {0} → Z \ {0},
2
x 7→ ⌊x⌋
is 2-to-1.
Exercise 4. Prove or disprove the following statements:
1. If a function f : S → T is injective and a function g : T → U is not injective, then
g ◦ f cannot be injective.
2. If a function f : S → T is surjective and a function g : T → U is not injective, then
g ◦ f cannot be injective.
1
Exercise 5. Let f1 : R → R and f2 : R → R be functions such that f1 is strictly monotonically increasing and f2 is strictly monotonically decreasing. Let f : R → R be defined
piecewise via
(
f1 (x) for x ≤ 0,
x 7→
f2 (x) for x > 0.
1. Show f is piecewise injective, namely f is injective on the subsets (−∞, 0] and
(0, ∞) of the domain R of f .
2. Is f also injective? Prove or disprove.
Exercise 6.
1. Find solutions to the following recurrence relations and initial conditions (without
proof ):
(a) an := 3an−1 , a0 := 2
(b) an := an−1 + n, a0 := 1
(c) an := 2an−1 − 1, a0 := 1
2. Compute the following sums:
P4
(a)
j=1 j!
P3 P3 k2
(b)
k=0 j!
j=1
Exercise 7. Prove the following statements:
1. The set of all closed intervals of finite length over the real numbers is not countable.
√
2. The set {x : x is rational or of the form p · 5 with p prime} is countable.
Hint: You may use statements like Cantor’s diagonal arguments from the lecture.
Exercise 8. Let fj denote the Fibonacci numbers. Prove by mathematical induction that
for all n ∈ N∗ it holds:
f1 + f3 + · · · + f2n−1 = f2n .
Exercise 9. Prove by mathematical induction that for all n ∈ N it holds:
n5 − n is divisible by 5.
You may use that (k + 1)5 = k 5 + 5k 4 + 10k 3 + 10k 2 + 5k + 1.
Exercise 10. Recall that for a number r ∈ N∗ we define rZ := {z ∈ Z : ∃x ∈ Z : z = r·x}.
2
1. Prove that for every r ∈ N∗ and n ∈ N∗ holds:
r n Z ⊆ rZ.
2. Show: p ∈ N∗ is in rZ if and only if r is a divisor of p.
Exercise 11. For a string w (over some given alphabet) let |w| denote its length and for
every 1 ≤ i, j ≤ |w| let w(i, j) denote the substring given by the i-th, (i + 1)-st,. . .,j-th
character in w. For two strings w1 and w2 we denote by w1 · w2 its concatenation.
Investigate the following pseudocode:
Input: A string w.
Output: A string w ′.
begin
if |w| = 1 then
return w
else
k
j
k
j
|w|+1
,
|w|
·
StringAlgo
w
1,
return StringAlgo w |w|+1
2
2
Algorithm 1: StringAlgo
1. Compute StringAlgo(“Hase”), StringAlgo(“Rabbit”) and StringAlgo(“Halloween”).
Describe in own words, what the algorithm does.
2. Show via strong induction: StringAlgo is injective.
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