In Exercises 1-10, determine whether the given relation is an equivalence relation on (1, 2, 3, 4, 5). If the relation is an equiv- alence relation, list the equivalence classes. (In Exercises 5-10, x, ye (1, 2, 3, 4, 5).) 1. {(1, 1), (2, 2), (3, 3). (4.4). (5, 5), (1,3), (3, 1))
In Exercises 1-10, determine whether the given relation is an equivalence relation on (1, 2, 3, 4, 5). If the relation is an equiv- alence relation, list the equivalence classes. (In Exercises 5-10, x, ye (1, 2, 3, 4, 5).) 1. {(1, 1), (2, 2), (3, 3). (4.4). (5, 5), (1,3), (3, 1))
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
INSTRUCTION: answer the numbers 1, 3, 7, 17, 20, 27, 30 only
![3.4 Exercises
In Exercises 1-10, determine whether the given relation is an
equivalence relation on {1, 2, 3, 4, 5). If the relation is an equiv-
alence relation, list the equivalence classes. (In Exercises 5-10,
x, ye (1, 2, 3, 4, 5).)
1. ((1, 1), (2, 2), (3, 3). (4,4). (5,5). (1,3), (3, 1))
2. {(1, 1), (2, 2), (3, 3), (4,4), (5,5), (1, 3), (3, 1), (3, 4). (4,3)}
3. ((1, 1), (2, 2), (3, 3), (4,4)}
4. {(1, 1), (2, 2), (3, 3), (4,4), (5, 5), (1, 5), (5. 1), (3, 5), (5,3).
(1,3), (3, 1))
5. ((x,y) | 1 ≤x≤ 5 and 1 ≤ y ≤ 5}
6. ((x, y) | 4 divides .x - y)
8. {(x, y) |x divides 2-y}
7. {(x, y) | 3 divides x + y}
9. {(x, y) |x and y are both even}
10. {(x, y) |x and y are both even or x and y are both odd}
In Exercises 11-16, determine whether the given relation is an
equivalence relation on the set of all people.
11. {(x, y) |x and y are the same height}
12. {(x, y) [x and y have, at some time, lived in the same country}
13. {(x, y) |x and y have the same first name}
14. ((x, y) |x is taller than y}
15. ((x, y) [x and y have the same parents}
16. ((x, y) [x and y have the same color hair}
In Exercises 17-22, list the members of the equivalence relation
on (1,2,3,4) defined (as in Theorem 3.4.1) by the given partition.
Also, find the equivalence classes [1], [2], [3], and [4].
17. ((1.2), (3, 4))
19. (1), (2), (3), (4))
21. (1,2,3,4}}
18. (1), (2), (3,4))
20. (1, 2, 3), (4}}
22.
(1), (2, 4), (3}}
In Exercises 23-25, let X=(1, 2, 3, 4, 5), Y=(3, 4), and C=
(1.3). Define the relation R on P(X), the set of all subsets of X.
as
ARB
ifAUY=BUY.
23. Show that R is an equivalence relation.
24. List the elements of [C], the equivalence class containing C.
25. How many distinct equivalence classes are there?
26. Let
X = (San Francisco, Pittsburgh, Chicago, San Diego,
Philadelphia, Los Angeles).
Define a relation R on X as x Ry if x and y are in the same state.
(a) Show that R is an equivalence relation.
(b) List the equivalence classes of X.
27. If an equivalence relation has only one equivalence class, what
must the relation look like?
28. If R is an equivalence relation on a finite set X and [X] = [R].
what must the relation look like?
29. By listing ordered pairs, give an example of an equivalence
relation on [1, 2, 3, 4, 5, 6) having exactly four equivalence
classes.
30. How many equivalence relations are there on the set (1, 2, 3)?
31. Let R be a reflexive relation on X satisfying: for all x, y, z € X,
if x Ry and y Rz, then z R.x. Prove that R is an equivalence re-
lation.
32. Define a relation R on RR, the set of functions from R to R.
byf Rg if f(0) = g(0). Prove that R is an equivalence relation
on RR. Let f(x) = x for all x € R. Describe [/].
33. Define a relation R on RR, the set of functions from R to R by
fRg if there exist a, b R such that f(x) = g(x + a) + b
for all x € R. Prove that R is an equivalence relation on
RR. What property do all functions in an equivalence class
share?
34. Let X = [1,2,.... 10). Define a relation R on X X X by
(a, b) R (c,d) if a+d=b+c.
(a) Show that R is an equivalence relation on X X X.
(b) List one member of each equivalence class of X X X.
35. Let X = (1,2,.... 10). Define a relation R on X X X by
(a, b) R (c,d) if ad=bc.
(a) Show that R is an equivalence relation on X X X.
(b) List one member of each equivalence class of X X X.
(c) Describe the relation R in familiar terms.
36. Let R be a reflexive and transitive relation on X. Show that
ROR is an equivalence relation on X.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6dc6da3d-9402-4f85-8a75-34c93da96fbe%2Fc6159057-021d-4461-b6c2-54c6f21fc187%2Fd6n91hl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3.4 Exercises
In Exercises 1-10, determine whether the given relation is an
equivalence relation on {1, 2, 3, 4, 5). If the relation is an equiv-
alence relation, list the equivalence classes. (In Exercises 5-10,
x, ye (1, 2, 3, 4, 5).)
1. ((1, 1), (2, 2), (3, 3). (4,4). (5,5). (1,3), (3, 1))
2. {(1, 1), (2, 2), (3, 3), (4,4), (5,5), (1, 3), (3, 1), (3, 4). (4,3)}
3. ((1, 1), (2, 2), (3, 3), (4,4)}
4. {(1, 1), (2, 2), (3, 3), (4,4), (5, 5), (1, 5), (5. 1), (3, 5), (5,3).
(1,3), (3, 1))
5. ((x,y) | 1 ≤x≤ 5 and 1 ≤ y ≤ 5}
6. ((x, y) | 4 divides .x - y)
8. {(x, y) |x divides 2-y}
7. {(x, y) | 3 divides x + y}
9. {(x, y) |x and y are both even}
10. {(x, y) |x and y are both even or x and y are both odd}
In Exercises 11-16, determine whether the given relation is an
equivalence relation on the set of all people.
11. {(x, y) |x and y are the same height}
12. {(x, y) [x and y have, at some time, lived in the same country}
13. {(x, y) |x and y have the same first name}
14. ((x, y) |x is taller than y}
15. ((x, y) [x and y have the same parents}
16. ((x, y) [x and y have the same color hair}
In Exercises 17-22, list the members of the equivalence relation
on (1,2,3,4) defined (as in Theorem 3.4.1) by the given partition.
Also, find the equivalence classes [1], [2], [3], and [4].
17. ((1.2), (3, 4))
19. (1), (2), (3), (4))
21. (1,2,3,4}}
18. (1), (2), (3,4))
20. (1, 2, 3), (4}}
22.
(1), (2, 4), (3}}
In Exercises 23-25, let X=(1, 2, 3, 4, 5), Y=(3, 4), and C=
(1.3). Define the relation R on P(X), the set of all subsets of X.
as
ARB
ifAUY=BUY.
23. Show that R is an equivalence relation.
24. List the elements of [C], the equivalence class containing C.
25. How many distinct equivalence classes are there?
26. Let
X = (San Francisco, Pittsburgh, Chicago, San Diego,
Philadelphia, Los Angeles).
Define a relation R on X as x Ry if x and y are in the same state.
(a) Show that R is an equivalence relation.
(b) List the equivalence classes of X.
27. If an equivalence relation has only one equivalence class, what
must the relation look like?
28. If R is an equivalence relation on a finite set X and [X] = [R].
what must the relation look like?
29. By listing ordered pairs, give an example of an equivalence
relation on [1, 2, 3, 4, 5, 6) having exactly four equivalence
classes.
30. How many equivalence relations are there on the set (1, 2, 3)?
31. Let R be a reflexive relation on X satisfying: for all x, y, z € X,
if x Ry and y Rz, then z R.x. Prove that R is an equivalence re-
lation.
32. Define a relation R on RR, the set of functions from R to R.
byf Rg if f(0) = g(0). Prove that R is an equivalence relation
on RR. Let f(x) = x for all x € R. Describe [/].
33. Define a relation R on RR, the set of functions from R to R by
fRg if there exist a, b R such that f(x) = g(x + a) + b
for all x € R. Prove that R is an equivalence relation on
RR. What property do all functions in an equivalence class
share?
34. Let X = [1,2,.... 10). Define a relation R on X X X by
(a, b) R (c,d) if a+d=b+c.
(a) Show that R is an equivalence relation on X X X.
(b) List one member of each equivalence class of X X X.
35. Let X = (1,2,.... 10). Define a relation R on X X X by
(a, b) R (c,d) if ad=bc.
(a) Show that R is an equivalence relation on X X X.
(b) List one member of each equivalence class of X X X.
(c) Describe the relation R in familiar terms.
36. Let R be a reflexive and transitive relation on X. Show that
ROR is an equivalence relation on X.
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