Recommended excercises
Discrete Mathemathics, 2023
Simón Isaza Peñaloza
This is a list of recommended excercises from the book of Kenneth Rosen
Discrete Mathematics and its applications listed by section. There are also some
excercises added by myself. I will be updating the list as the course progresses.
It is convenient that the students additionally solve other excercises from
the book other than those listed here: The excercises proposed here should be
considered a minimum.
Module 1: Propositional Logic, Predicate Logic and Proofs.
Section 1.1: Exercises 1-7, 10, 13, 14, 17, 18, 19, 21, 24, 29, 31, 33, 39, 46, 47.
(i) In excercises 10, 13, 14 and 17, assume p is true, q is false, and r is true.
Find the truth values of the listed propositions.
Section 1.2: Exercises 44, 45.
Section 1.3: Exercises 5, 6, 9, 12, 14, 17, 18, 20, 21, 26, 27.
Section 1.4: Exercises 1, 2, 4, 5, 7-14, 28, 32, 33, 35, 55.
Section 1.5: Exercises 1, 4, 8, 9, 23, 28, 30, 38, 42, 45.
Section 1.6:
(i) For each of the following arguments, if it is valid, give a derivation (Indicate
the inference rule used on each step), and if it is not valid, show why.
a)
P ∧Q
(P ∨ Q) → R
R
c)
E→F
¬G → ¬F
H→I
E∨H
G∨I
b)
¬X → Y
¬X → Z
¬Z → ¬Y
d)
L→M
(M ∨ N ) → (L → K)
¬P ∧ L
K
1
e)
P →Q
¬R → (S → T )
R ∨ (P ∨ T )
¬R
Q∨S
¬A → (B → ¬C)
C → ¬A
(¬D ∨ A) → ¬¬C
¬D
¬B
f)
(ii) For each of the following arguments, if it is valid, give a derivation, and if
it is not valid, show why.
a) If Fishville is boring, then it is hard to find. If Fishville is not small,
then it is not hard to find. Fishville is boring. Therefore Fishville is
small.
b) If the food is green, then it is undercooked. If the food is smelly,
then it is stale. The food is green or it is stale. Therefore the food is
undercooked or it is smelly.
c) If you rob a bank, you go to jail. If you go to jail, you do not have
fun. If you have a vacation, you have fun. You rob a bank or you
have a vacation. Therefore you go to jail or you have fun.
d) It is not the case that Fred plays both guitar and flute. If fred does
not play guitar and he does not not play flute, then he plays both
organ and harp. If he plays harp, then he plays organ. therefore Fred
plays organ.
Section 1.7:
For all the proof excercises you should always write complete proofs with
full sentences. While you may discuss the ideas behind the proofs, you won’t
learn by transcribing your classmate’s proofs: You should, at the very least, be
capable of writing the proof on your own.
Exercises 16, 17, 19.
Definition. Let a and b be integers. We say that a divides b if there is an
integer q such that aq = b. We write a|b and say that a is a factor of b and that
b is divisible by a. If a does not divide b we write a ∤ b.
(i) Let a, b, c, n and m be integers. Prove the following statements.
a) Any integer divides zero.
b) If a|b and b|c then a|c.
c) If a|b and a|c then a|bm + cn.
d) If a|b, then ak |bk for every k ∈ Z.
e) If a ∤ bc then a ∤ b.
f) Suppose there is some d ∈ Z such that d|a and d|b but d ∤ c. Then
no integers x and y satisfy the equation ax + by = c.
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g) Let a, b ̸= 0. Then a|b and b|a if and only if a = b or a = −b.
(ii) Let m and n be integers. Prove that mn is odd iff both m and n are odd.
a b
(iii) Let M =
be an upper triangular2 × 2 integer matrix. Prove that
0 d
the following are equivalent:
(1) det(A) = 1
(2) a = d = ±1
(3) a + d = ±2 and a = d
(To prove that more than two statements are equivalent it suffices to prove
the implications ciclically. In this case it’s enough to prove (1) ⇒ (2),
(2) ⇒ (3) and (3) ⇒ (1). What rule of inference makes this sufficient?
Syllogism).
Definition. Let a and b be integers. We say that a and b are relatively prime
it they have no common divisor different from 1 and −1.
(iv) Prove that There exist integers a and b that are relatively prime and there
exist integers a and b that are not relatively prime.
(v) Prove that the following statement is false: If two integers are relatively
prime then they are both even.
More challenging excercises:
(vi) Prove that the only consecutive non-negative integers a, b and c that
satisfy a2 + b2 = c2 are 3, 4 and 5.
(vii) Prove that there are infinitely many prime numbers. (Hint: By absurd.
Suppose there are finitely many prime numbers. Show that the product
of all the prime numbers plus one is prime. Use the fact that a natural
number greater that 1 has a prime factor).
Module 2: Sets
Section 2.1: Excercise 45.
Section 2.2: Excercises 1-4, 14, 16, 28, 29, 31.
(i) Determine which of the following are true or false.
3
a) 3 ∈ (3, 5]
b) 10 ∈
/ −∞, π
f) [1, 2] ⊆ 0, 1, 2, 3
2
g) {−1, 0, 1} ⊆ [1, 2)
c) 7 ∈ {2, 3, . . . , 11}
h) [5, 7] ⊆ (4, ∞)
d) π ∈ (2, ∞)
i) {2, 4, 8, 16, . . . } ⊆ [2, ∞)
e) −1.3 ∈ {. . . , −3, −2, −1}
j) ∅ ⊆ [2, ∞)
(ii) Determine which of the following are true or false.
a)
b)
c)
d)
e)
f)
g)
h) ∅ ⊆ {∅}
0∈∅
{0} ⊆ ∅
∅ ⊆ {0}
{0} ⊆ {0}
{0} ∈ {0}
∅ ∈ {∅}
∅⊆∅
i) {∅} ⊆ ∅
j) {∅} ⊆ {∅}
k) {∅} ⊆ {{∅}}
l) {∅} ∈ {{∅}}
m) {∅} ⊆ {∅, {∅}}
(iii) Let P be the set off all people, F the set of all women and M the set of
all men. Describe each of the following sets with words.
a) {x ∈ P | x ∈ M and x has a child}
b) {x ∈ P | ∃y, z ∈ P such that y is a child of x and z is a child of y}
c) {x ∈ P | ∃m ∈ F such that x is married to m}
d) {x ∈ P | ∃n ∈ M such that x is the child of n and x is older than n}
(iv) Describe the set {x ∈ R | x > 0 ∧ x2 > 1} in words, as an interval and as
an intersection of two sets.
(v) Describe the set {x ∈ R | x ∈ Z ∨ x > π} in words, as a union of two sets
and as a difference of two sets.
(vi) Describe the set {f : R −→ R | ∀x (f (x) > 0 ∨ f (x) = f (−x))} in words,
as a union of two sets and as a difference of two sets.
(vii) Describe formally (without words, in the fashion {x | P (x)} ) the following
sets.
a) The set of all possitive real numbers.
b) The set of all odd integers.
c) The set of all rational numbers that have a factor 5 in their denominator.
(viii) In excercise 3 from Section 2.2, find also A × B and P(A).
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Section 2.3: Excercises 3, 6*, 10, 11, 20, 35.
* Remark: The statement of the problem 6 is informal. A well defined function must have a domain, codomain, and assignation defined from the beginning,
so “Find the domain and range of these functions” is an informal statement,
though quite frequent. What this means in proper terms is: ”On each case a),
b), c), etc., find a departure set A and an arriving set B such that we can define
a function f : A → B that assigns elements of B to elements of A as described,
and such that its range is B.
(i) Let X denote the set of all people, Which of the following descriptions
define functions from X to X?
a)
b)
c)
d)
e)
f (x) is x’s mother.
g(x) is x’s brother.
h(x) is x’s friend (assume everybody has friends).
j(x) is x’s best friend (assume everybody has one).
k(x) is x’s first born child if he or she is a parent, and his or her
father otherwise.
f) If Y is the set of countries, n : X → Y , n(x) is x’s nationality.
(ii) For each of the previous cases found to be functions, indicate if the given
function is injective, surjective, bijective neither.
(iii) For each of the following cases indicate if the given function is injective,
surjective, bijective, neither, or not a well defined function.
a)
b)
c)
d)
t : (1, ∞) → R, t(x) = ln x.
s : (−∞, 0) → (−∞, 0), s(x) = x3 .
h : R2 → R, h((x, y)) = x2 + y 2 .
Q : N → P(N), Q(n) = [[0, n]].
(iv) For each of the following cases indicate if the given function is injective,
surjective, bijective, neither, or not a well defined function.
a)
b)
c)
d)
e)
f : R → R, f (x) = x2 + 1.
g : R → [1, ∞), g(x) = x2 + 1.
h : R → [1, 2), h(x) = x2 + 1.
j : [0, ∞) → R, j(x) = x2 + 1.
k : [0, ∞) → [1, ∞), k(x) = x2 + 1.
(v) For each pair of functions given below, find formulas for f ◦ g and g ◦ f
(simplifying when possible).
a)
b)
c)
d)
f, g : R → R, f (x) = x2 − 1, g(x) = x2 − 1. Find also g ◦ g.
f, g : R → R, f (x) = ex , g(x) = sin(x).
f, g : R+ → R+ , f (x) = x7 , g(x) = x−3 .
√
f : R → [0, ∞), f (x) = x6 ; g(x) : [0, ∞) → R, g(x) = 6 x.
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Module 3: Recursion and Arithmetic
Section 2.4: Examples 12, 15 (To be made as exercises); Exercises 1, 2, 9, 14,
16, 19, 20, 29, 33.
Section 5.1: Examples 1-4, 8, 11 (To be made as exercises); Exercises 3, 9, 18,
38, 56.
Section 5.2: Example 3 (To be made as an exercise); Exercises 1, 7.
Section 5.3: Exercises 1, 5, 7, 8, 31, 36, 37, 39, 65.
Section 5.4: Example 2 (To be made as an exercise); Exercises 1, 7, 8, 9, 10, 23.
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