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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

Prove your assertions.

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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