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# ДискретнаяМатематика / Student Solutions Manual / chapter 3

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3.3 Exercises

1.For the people in the family tree (see Figure 3-2), build tables for the following relations:

(a)IsAncestorOf

(b)IsDescendentOf

(c)IsSiblingOf

(d)IsCousinOf

1.

Mary = John

Peter=Elaine Maude=Harold

 George Elizabeth

1. (a)

IsAncestorOf

 Elaine George Mary George John George Peter George Harold Elizabeth Maude Elizabeth Mary Elizabeth John Elizabeth Mary Elaine John Elaine Mary Maude John Maude

1. (b)

IsDescendentOf

 George Mary George John George Peter George Elaine Maude Mary Maude John Elizabeth Maude Elizabeth Harold Elizabeth Mary Elizabeth John Elaine Mary Elaine John
 1. (c) IsSiblingOf Elaine Maude Maude Elaine 1. (d) IsCousinOf George Elizabeth Elizabeth George

3.Find the elements in each of the following relations de ned on R:

(a)(x; y) 2 R if and only if x + 1 < y

(b)(x; y) 2 R if and only if y < 0 or 2 x 3

(c)(x; y; z) 2 R if and only if x2 + y = z

3. (a)

3. (b)

3. (c)

10

 5 2

0

0

-2

0

-2

2

5. The table gives the names of airlines and several cities that each ies to from Chicago. The table also gives the number of miles for each ight. List all the triples (X, Y, Z) of the ternary relation de ned by those triples for which airline X ies Y miles to city Z.

 TWA Pan Am Piedmont Topeka 603 Bombay 7809 Peoria 170 Kansas City 510 Seattle 2052 Albany 816 Phoenix 1742 Anaheim 2025 Atlanta 717 5. Airlines Mileage City TWA 603 Topeka TWA 510 Kansas City TWA 1742 Phoenix Pan Am 7809 Bombay Pan Am 2052 Seattle Pan Am 2025 Anaheim Piedmont 170 Peoria Piedmont 816 Albany Piedmont 717 Atlanta

7.Using the family tree shown in Figure 3-2, list the elements in each of the following relations, and give these relations meaningful names.

(a)IsMarriedTo 1

(b)IsMarriedTo IsMarriedTo

(c)IsParentOf IsParentOf 1

(d)=Family where Family denotes the set of people appearing in the family tree

 (e) IsMarriedTo \ IsMarriedTo 1 (f) IsParentOf \ IsParentOf 1 7. (a) IsMarriedTo 1 John Mary Mary John Peter Elaine Elaine Peter Maude Harold Harold Maude

7. (b)

IsMarriedTo IsMarriedTo

Peter Peter

Elaine Elaine

Maude Maude

Harold Harold

Mary Mary

John John

7. (c)

IsParentOf IsParentOf 1

 Elaine Elaine Elaine Maude Maude Elaine Maude Maude George George Elizabeth Elizabeth

7. (d)

=Family

 Peter Peter Elaine Elaine Mary Mary John John Maude Maude Harold Harold Elizabeth Elizabeth George George

7. (e)

IsMarriedTo \ IsMarriedTo 1

John Mary

Mary John

Peter Elaine

Elaine Peter

Maude Harold

Harold Maude

7.(f) ;

9.Prove Theorem 2(c).

9.For any x; y 2 X such that (x; y) 2 S 1, it follows that (y; x) 2 S: Since S R; (y; x) 2 R: Now, (x; y) 2 R 1:

11.Let A = f1; 2; 3; : : : ; 10g: Let R = f(1, 2), (1, 4), (1, 6), (1, 8), (1, 10), (3, 5), (3, 7), (4, 6), (6, 8), (7, 10)g be a relation on A: Let S = f(2; 4); (3; 6); (5; 7); (7; 9); (8; 10); (8; 9); (8; 8); (9; 9); (3; 8); (4; 9)g be a second relation on A: Find:

(a)R S

(b)S R:

11. (a) f(2; 6); (3; 8); (5; 10)g

11. (b) f(1; 4); (1; 9); (1; 10); (1; 8); (3; 7); (3; 9); (6; 10); (6; 9); (6; 8)g

13.Let R; S and T be binary relations on a set X.

(a)Prove that R S if and only if R 1 S 1:

(b)Prove that if R S, then R T S T and T R T S:

(c)If R T S T and T R T S for some relation T , does it follow that R S?

(d)If R T S T and T R T S for every relation T , does it follow that R S?

Prove that your answers to (c) and (d) are correct. 13. (a) Assume that R S.

()) Let (x; y) 2 R 1:

(x; y) 2 R 1 ) (y; x) 2 R ) (y; x) 2 S ) (x; y) 2 S 1

Therefore, R 1 S 1.

(() Now assume that R 1 S 1. Then, by the result above,

(R 1) 1 (S 1) 1)

Thus, by Theorem 2, R S.

13. (b)

 (x; y) 2 R T ) there is a z such that (z; y) 2 R and (x; z) 2 T ) (z; y) 2 S and (x; z) 2 T ( since R S) ) (x; y) 2 S T Therefore, if R S, then R T S T: (x; y) 2 T R ) there is a x such that (z; y) 2 T and (x; z) 2 R ) (z; y) 2 T and (x; z) 2 S( since R S) ) (x; y) 2 T S

Therefore, if R S, then T R T S:

13. (c) No. Note that for any relations R, S on any set X, R; = ; = S ;. But it does not follow that R S, for example, let X = N; R = LTN; S =

;.

13. (d) Yes. Take T = IdX : By hypothesis, R IdX S IdX {that is,

R S.

3.5 Exercises

1.Which of the following relation on the set of all people are re exive? Symmetric? Antisymmetric? Transitive? Prove your assertions.

(a)R(x; y) if y makes more money than x.

(b)R(x; y) if x and y are about the same height.

(c)R(x; y) if x and y have an ancestor in common.

(d)R(x; y) if x and y are the same sex.

(e)R(x; y) if x and y both collect stamps.

(f) R(x; y) if x and y like some of the same music.

1.

 Re exive Symmetric Antisymmetric Transitive (a) No No Yes Yes (b) Yes Yes No Yes (c) Yes Yes No No (d) Yes Yes No Yes (e) Yes Yes No Yes (f) Yes Yes No No

3.Which of the following relations on the set of all people are re exive? Symmetric? Antisymmetric? Transitive? Explain why your assertions are true.

(a)R(x; y) if x and y either both like German food or both dislike German food.

(b)R(x; y) if (i) x and y either both like Italian food or both dislike it, or

(ii)x and y either both like Chinese food or both dislike it.

(c) R(x; y) if y is at least two feet taller than x:

3.

 Re exive Symmetric Antisymmetric Transitive (a) Yes Yes No Yes (b) Yes Yes No No (c) No No Yes Yes

5.Which of the following relations on the set of people indicated are re exive? Irre exive? Symmetric? Antisymmetric? Transitive?

(a)IsSisterOf on the set of all females

(b)IsBrotherOfOrEquals on the set of all males

(c)IsSiblingOf on the set of all people

(d)IsSiblingOfOrEquals on the set of all people

(e)IsCousinOfOrEquals on the set of all people

5.

 Re ex. Irre ex. Symm. Antisymm. Trans. (a) IsSisterOf N Y Y N N (b) IsBrotherOfOrEquals Y N Y N Y (c) IsSiblingOf N Y Y N N (d) IsSiblingOfOrEquals Y N Y N Y (e) IsCousinOfOrEquals Y N Y N N

7. Let A = fa; b; c; dg: De ne the relations R1 and R2 on A as

R1 = f(a; a); (a; b); (b; d)g

and

R2 = f(a; d); (b; c); (b; d); (c; b)g

Find

(a)R1 R2

(b)R2 R1

(c)R12

(d)R22

7. (a) (c, d)

7.(b) (a, d), (a, c)

7.(c) (a, a), (a, b), (a, d)g

7. (d) f(b, b), (c, d), (c, c)g

9.In the example involving the family tree (see Figure 3-2),

(a)What is the transitive and re exive closure of IsParentOf?

(b)What is the transitive and re exive closure of IsMarriedTo?

9. In the answers only the transitive closure is shown. To include the re exive closure you must add all the pairs of the form (x; x) where x is a person in the family tree.

9. (a)

Mary = John

Peter=Elaine Maude=Harold

 George Elizabeth Transitive Closure of IsParentOf Elaine George Peter George Mary Elaine John Elaine Maude Elizabeth Harold Elizabeth Mary Maude John Maude Mary George John George Mary Elizabeth John Elizabeth

9. (b)

Transitive Closure of IsMarriedTo

Mary John

John Mary

Peter Elaine

Elaine Peter

Maude Harold

Harold Maude

11. Let X = f1; 2; 3; 4g, and de ne a relation R on X as

R= f(1; 2); (2; 3); (3; 4)g

(a)Find the re exive closure of R:

(b)Find the symmetric closure of R:

(c)Find the transitive closure of R:

(d)Find the re exive and transitive closure of R:

11. (a) R [ IdX = f(1, 2), (2, 3), (3, 4), (1, 1), (2, 2), (3, 3), (4, 4)g

11. (b) R [ R 1 = f(1, 2), (2, 3), (3, 4), (2, 1), (3, 2), (4, 3)g

11. (c) R+ = f(1, 2), (2, 3), (3, 4), (1, 3), (2, 4), (1, 4) g

11.(d) R = R+[ f(1, 1), (2, 2), (3, 3), (4, 4)g

13.Let A = f1, 2, 3, 4g. Find the transitive closure of the relation R de ned on A as

R= f(1; 2); (2; 1); (2; 3); (3; 4)g

13.f(1; 2); (2; 1); (2; 3); (3; 4); (1; 1); (1; 3); (2; 2); (2; 4); (1; 4)g

15.Let X = f4, 5, 6, 7, 8g, and de ne the relation R on X as f(4, 5), (5, 6), (6, 7), (7, 8), (8, 4)g. Find the smallest integers m and n such that

Rm = Rn where 0 < m < n:

15.m = 1 and n = 6:

17.Show that the transitive closure of a relation R on a set X is the intersection of all transitive binary relations R0 on X where R R0.

17.Denote by T the intersection of all transitive binary relations containing R. It follows from Exercise 19 (c) that R+ T , and it follows from Exercise 19 (b) that T R+. So, R+ = T .

19.Prove Theorem 5(c) as follows:

(a)Prove by induction that if R is a binary relation on a set X, then

Rm Rn = Rm+n where m; n 2 N:

(b)Prove that R+ is transitive.

(c)Prove by induction that if R S and S is a transitive binary relation, then Rn S: Conclude that R+ S:

19. (a) Let m be arbitrary. If n = 0; then

Rm Rn = Rm R0 = Rm = Rm+0 = Rm+n

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