tomit4/discrete_mathematics_with_applications
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

🚧 Setup for 8.2

Browse source
This commit is contained in:
tomit4 2026-08-15 17:52:22 -07:00
parent 08d5628616
commit 08a707caa7
3 changed files with 323 additions and 0 deletions
Download patch file Download diff file Expand all files Collapse all files

222
chapter_8/exercises.md
View file

@ -564,3 +564,225 @@ Primary_Diagnosis = appendicitis.
(466581, Mary Lazars),
(778400, Jamal Baskers)
---
Page 526
**Exercise Set 8.2**
In 1-8, a number of relations are defined on the set $A = \{0, 1, 2, 3\}$. For
each relation:
a. Draw the directed graph.
b. Determine whether the relation is reflexive.
c. Determine whether the relation is symmetric.
d. Determine whether the relation is transitive.
Give a counterexample in each case in which the relation does not satisfy one of
the properties.
1. $R_1 = \{(0, 0), (0, 1), (0, 3), (1, 1), (1, 0), (2, 3), (3, 3)\}$
2. $R_2$ = \{(0, 0), (0, 1), (1, 1), (1, 2), (2, 2), (2, 3)\}
3. $R_3 = \{(2, 3), (3, 2)\}$
4. $R_4 = \{(1, 2), (2, 1), (1, 3), (3, 1)\}$
5. $R_5 = \{(0, 0), (0, 1), (0, 2), (1, 2)\}$
6. $R_6 = \{(0, 1), (0, 2)\}$
7. $R_7 = \{(0, 3), (2, 3)\}$
8. $R_8 = \{(0, 0), (1, 1)\}$
In 9-33, determine whether the given relation is reflexive, symmetric,
transitive, or none of these. Justify your answers.
9. $R$ is the "greater than or equal to" relation on the set of real numbers:
For every $x, y \in \mathbb{R}$, $x R y \Leftrightarrow x \geq y$.
10. $C$ is the circle relation on the set of real numbers: For every
$x, y \in \mathbb{R}, x C y \Leftrightarrow x^2 + y^2 = 1$.
11. $D$ is the relation defined on $\mathbb{R}$ as follows: For every
$x, y \in \mathbb{R}, x D y \Leftrightarrow xy \geq 0$.
12. $E$ is the congruence modulo $4$ relation on $\mathbb{Z}$: For every
$m, n \in \mathbb{Z}, m E n \Leftrightarrow 4 | (m - n)$.
13. $F$ is the congruence modulo $5$ relation on $\mathbb{Z}$: For every
$m, n \in \mathbb{Z}, m F n \Leftrightarrow 5 | (m - n)$.
14. $O$ is the relation defined on $\mathbb{Z}$ as follows: For every
$m, n \in \mathbb{Z}, m O n \Leftrightarrow m - n \text{ is odd}$.
15. $D$ is the "divides" relation on $\mathbb{Z}^+$: For all positive integers
$m$ and $n$, $m D n \Leftrightarrow m | n$.
16. $A$ is the "absolute value" relation on $\mathbb{R}$: For all real numbers
$x$ and $y$, $x A y \Leftrightarrow |x| = |y|$.
17. Recall that a prime number is an integer that is greater than $1$ and has no
positive integer divisors other than $1$ and itself. (In particular, $1$ is
not prime.) A relation $P$ is defined on $\mathbb{Z}$ as follows: For every
$m, n \in \mathbb{Z}, m P n \Leftrightarrow \exists \text{ a prime number } p \text{ such that } p | m \text{ and } p | n$.
18. Define a relation $Q$ on $\mathbb{R}$ as follows: For all real numbers $x$
and $y$, $x Q y \Leftrightarrow x - y$ is rational.
19. Define a relation $I$ on $\mathbb{R}$ as follows: For all real numbers $x$
and $y$, $x I y \Leftrightarrow x - y$ is irrational.
20. Let $X = \{a, b, c\}$ and $\mathscr{P}(X)$ be the power set of $X$ (the set
of all subsets of $X$). A relation $\mathbf{E}$ is defined on
$\mathscr{P}(X)$ as follows: For every
$A, B \in \mathscr{P}(X), A \mathbf{E} B \Leftrightarrow \text{ the number of elements in } A \text{ equals the number of elements in } B$.
21. Let $X = \{a, b, c\}$ and $\mathscr{P}(X)$ be the power set of $X$. A
relation $\mathbf{L}$ is defined on $\mathscr{P}(X)$ as follows: For every
$A, B \in \mathscr{P}(X), A \mathbf{L} B \Leftrightarrow \text{ the number of elements in } A \text{ is less than the number of elements in } B$.
22. Let $X = \{a, b, c\}$ and $\mathscr{P}(X)$ be the power set of $X$. A
relation $\mathbf{N}$ is defined on $\mathscr{P}(X)$ as follows: For every
$A, B \in \mathscr{P}(X), A \mathbf{N} B \Leftrightarrow \text{ the number of elements in } A \text{ is not equal to the number of elements in } B$.
23. Let $X$ be a nonempty set and $\mathscr{P}(X)$ the power set of $X$. Define
the "subset" relation $\mathbf{S}$ on $\mathscr{P}(X)$ as follows: For every
$A, B \in \mathscr{P}(X), A \mathbf{S} B \Leftrightarrow A \subseteq B$.
24. Let $X$ be a nonempty set and $\mathscr{P}(X)$ the power set of $X$. Define
the "not equal to" relation $\mathbf{U}$ on $\mathscr{P}(X)$ as follows: For
every $A, B \in \mathscr{P}(X), A \mathbf{U} B \Leftrightarrow A \neq B$.
25. Let $A$ be the set of all strings of _a_'s and _b_'s of length $4$. Define a
relation $R$ on $A$ as follows: For every
$s, t \in A, s R t \Leftrightarrow s \text{ has the same first two characters as } t$.
26. Let $A$ be the set of all strings of 0's, 1's, and 2's that have length 4
and for which the sum of the characters in the string is less than or equal
to 2. Define a relation $R$ on $A$ as follows: For every
$s, t \in A, s R t \Leftrightarrow \text{ the sum of the characters of } s \text{ equals the sum of the characters of } t$.
27. Let $A$ be the set of all English statements. A relation $\mathbf{I}$ is
defined on $A$ as follows: For every $p, q \in A$,
$$ p \mathbf{I} q \Leftrightarrow p \to q \text{ is true} $$
28. Let $A = \mathbb{R} \times \mathbb{R}$. A relation $\mathbf{S}$ is defined
on $A$ as follows: For every $(x_1, y_1)$ and $(x_2, y_2)$ in $A$,
$$ (x_1, y_2) \mathbf{S} (x_2, y_2) \Leftrightarrow x_1 = x_2 $$
29. Let $A = \mathbb{R} \times \mathbb{R}$. A relation $\mathbf{S}$ is defined
on $A$ as follows: For every $(x_1, y_1)$ and $(x_2, y_2)$ in $A$,
$$ (x_1, y_2) \mathbf{S} (x_2, y_2) \Leftrightarrow y_1 = y_2 $$
30. Let $A$ be the "punctured plane"; that is, $A$ is the set of all points in
the Cartesian plane except the origin $(0, 0)$. A relation $R$ is defined on
$A$ as follows: For every $p_1$ and $p_2$ in $A$,
$p_1 R p_2 \Leftrightarrow p_1 \text{ and } p_2 \text{ lie on the same half line emanating from the origin}$.
31. Let $A$ be the set of people living in the world today. A relation $R$ is
defined on $A$ as follows: For all people $p$ and $q$ in $A$,
$$ p R q \Leftrightarrow p \text{ lives within 100 miles of } q $$
32. Let $A$ be the set of all lines in the plane. A relation $R$ is defined on
$A$ as follows: For every $l_1$ and $l_2$ in $A$,
$l_1 R l_2 \Leftrightarrow l_1 \text{ is parallel to } l_2$. (Assume that a
line is parallel to itself.)
33. Let $A$ be the set of all lines in the plane. A relation $R$ is defined on
$A$ as follows: For every $l_1$ and $l_2$ in $A$,
$$ l_1 R l_2 \Leftrightarrow l_1 \text{ is perpendicular to } l_2 $$
In 34-36, assume that $R$ is a relation on a set $A$. Prove or disprove each
statement.
34. If $R$ is reflexive, then $R^{-1}$ is reflexive.
35. If $R$ is symmetric, then $R^{-1}$ is symmetric.
36. If $R$ is transitive, then $R^{-1}$ is transitive.
In 37-42, assume that $R$ and $S$ are relations on a set $A$. Prove or disprove
each statement.
37. If $R$ and $S$ are reflexive, is $R \cap S$ reflexive? Why?
38. If $R$ and $S$ are symmetric, is $R \cap S$ symmetric? Why?
39. If $R$ and $S$ are transitive, is $R \cap S$ transitive? Why?
40. If $R$ and $S$ are reflexive, is $R \cup S$ reflexive? Why?
41. If $R$ and $S$ are symmetric, is $R \cup S$ symmetric? Why?
42. If $R$ and $S$ are transitive, is $R \cup S$ transitive? Why?
In 43-50, the following definitions are used: A relation on a set $A$ is defined
to be
irreflexive if, and only if, for every $x \in A, x \cancel{R} x$;
asymmetric if, and only if, for every $x, y \in A$ if $x R y$ then
$y \cancel{R} x$;
intransitive if, and only if, for every $x, y, z \in A$, if $x R y$ and $y R z$
then $x \cancel{R} z$.
For each of the relations in the referenced exercise, determine whether the
relation is irreflexive, asymmetric, intransitive, or none of these.
43. Exercise 1
44. Exercise 2
45. Exercise 3
46. Exercise 4
47. Exercise 5
48. Exercise 6
49. Exercise 7
50. Exercise 8
In 51-53, $R$, $S$, and $T$ are relations defined on $A = \{0, 1, 2, 3\}$.
51. Let $R = \{(0, 1), (0, 2), (1, 1), (1, 3), (2, 2), (3, 0)\}$.
Find $R^t$, the transitive closure of $R$.
52. Let $S = \{(0, 0), (0, 3), (1, 0), (1, 2), (2, 0), (3, 2)\}$.
Find $S^t$, the transitive closure of $S$.
53. Let $T = \{(0, 2), (1, 0), (2, 3), (3, 1)\}$.
Find $T^t$, the transitive closure of $T$.
54. Write a computer algorithm to test whether a relation $R$ defined on a
finite set $A$ is reflexive, where
$$ A = \{a[1], a[2], \dots, a[n]\} $$
55. Write a computer algorithm to test whether a relation $R$ defined on a
finite set $A$ is symmetric, where
$$ A = \{a[1], a[2], \dots, a[n]\} $$
56. Write a computer algorithm to test whether a relation $R$ defined on a
finite set $A$ is transitive, where
$$ A = \{a[1], a[2], \dots, a[n]\} $$