COMP232 - Mathematics for Computer Science

Transcription

1 COMP232 - Mathematics for Computer Science Tutorial 11 Ali Moallemi moa ali@encs.concordia.ca Iraj Hedayati h iraj@encs.concordia.ca Concordia University, Fall 2015 Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 1 / 1

2 Table of Contents Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 2 / 1

3 Exercise 3 For each of these relations on the set {1, 2, 3, 4}, decide whether it is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive. a) {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)} Reflexive: NO (1, 1) R Symmetric: NO (2, 4) R but (4, 2) R Antisymmetric: NO (2, 3) R and (3, 2) R where 2 3 Transitive: YES b) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)} Reflexive: YES Symmetric: YES Antisymmetric: NO (1, 2) R and (2, 1) R where 1 2 Transitive: YES Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 3 / 1

4 Exercise 3 - Cont... For each of these relations on the set {1, 2, 3, 4}, decide whether it is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive. c) {(2, 4), (4, 2)} Reflexive: NO (2, 2) R Symmetric: YES Antisymmetric: NO (2, 4) R and (4, 2) R where 2 4 Transitive: NO (2, 4), (4, 2) R but (2, 2) R d) {(1, 2), (2, 3), (3, 4)} Reflexive: NO (1, 1) R Symmetric: NO (2, 3) R but (3, 2) R Antisymmetric: YES Transitive: NO (2, 3), (3, 4) R but (2, 4) R Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 4 / 1

5 Exercise 3 - Cont... For each of these relations on the set {1, 2, 3, 4}, decide whether it is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive. e) {(1, 1), (2, 2), (3, 3), (4, 4)} Reflexive: YES Symmetric: YES Antisymmetric: YES Transitive: YES f) {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)} Reflexive: NO (1, 1) R Symmetric: NO (1, 4) R but (4, 1) R Antisymmetric: NO (1, 3), (3, 1) R but 1 3 Transitive: NO (2, 3), (3, 1) R but (2, 1) R Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 5 / 1

6 Exercise 5 Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) R if and only if a) everyone who has visited Web page a has also visited Web page b Reflexive: YES Symmetric: NO Antisymmetric: NO Transitive: YES b) there are no common links found on both Web page a and Web page b. Reflexive: NO Symmetric: YES Antisymmetric: NO Transitive: NO Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 6 / 1

7 Exercise 5 - Cont... Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) R if and only if c) there is at least one common link on Web page a and Web page b. Reflexive: NO Symmetric: YES Antisymmetric: NO Transitive: NO d) there is a Web page that includes links to both Web page a and Web page b. Reflexive: NO Symmetric: YES Antisymmetric: NO Transitive: NO Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 7 / 1

8 Exercise 6 Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) R if and only if a) x + y = 0 Reflexive: NO = 2 Symmetric: YES x + y = 0 y + x = 0 Antisymmetric: NO x + y = 0 y + x = 0 x = y x y Transitive: NO (x, y), (y, z) R and y 0, then x + y = 0 and y + z = 0 implies x + 2y + z = 0. Hence x + z can not be 0. Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 8 / 1

9 Exercise 6 - Cont... Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) R if and only if b) x = ±y Reflexive: NO x x Symmetric: YES x = ±y y = ±x Antisymmetric: NO x = ±y and y = ±x, x y because x x Transitive: YES Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 9 / 1

10 Exercise 6 - Cont... Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) R if and only if c) x y is a rational number Reflexive: YES x x = 0 Q Symmetric: YES x y Q y x = (x y) Q Antisymmetric: NO 2 1 = = 1 but 2 1 Transitive: YES Let a = x y and b = y z. Then c = a + b = x y + y z = x z. As a, b Q, then c Q and (x, z) R Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 10 / 1

11 Exercise 6 - Cont... Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) R if and only if d) x = 2y Reflexive: NO x 2x Symmetric: NO x = 2y y = x 2 2x Antisymmetric: YES x = 2y and y = 2x x = 4x x = 0 x = 2x = y Transitive: NO x = 2y and y = 2z then x = 4z Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 11 / 1

12 Exercise 6 - Cont... Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) R if and only if e) xy 0 Reflexive: YES x x = x 2 0 Symmetric: YES xy 0 yx 0 Antisymmetric: NO x = 1 and y = 2. (x, y), (y, x) R but x y Transitive: NO x = 1, y = 0, z = 1 then (x, y) R and (y, z) R but (x, z) R Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 12 / 1

13 Exercise 6 - Cont... Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) R if and only if f) xy = 0 Reflexive: NO x = 1 Symmetric: YES xy = 0 yx = 0 Antisymmetric: NO x = 1 and y = 0. (x, y), (y, x) R but x y Transitive: NO x = 1, y = 0 and z = 2. Then (x, y) R and (y, z) R but (x, z) R Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 13 / 1

14 Exercise 6 - Cont... Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) R if and only if g) x = 1 Reflexive: NO x = 2 (2, 2) R Symmetric: NO x = 1 y R (x, y) R but y R {1} (y, x) R. For example y = 2: (1, 2) R and (2, 1) R Antisymmetric: YES (x, y) R x = 1 and also (y, x) R y = 1. Hence x = y = 1 Transitive: YES (x, y) R x = 1 and also (y, z) R y = 1. Hence (x, z) R Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 14 / 1

15 Exercise 6 - Cont... Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) R if and only if h) x = 1 or y = 1 Reflexive: NO x = 2 (2, 2) R Symmetric: YES (x, y) R x = 1 and also (y, x) R Antisymmetric: NO x = 1 and y = 2. Then (x, y), (y, x) R but x y Transitive: NO x = 2, y = 1 and z = 3. Then (x, y) R and (y, z) R but (x, z) R Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 15 / 1

16 Exercise 31 Let A be the set of students at your school and B the set of books in the school library. Let R 1 and R 2 be the relations consisting of all ordered pairs (a, b), where student a is required to read book b in a course, and where student a has read book b, respectively. Describe the ordered pairs in each of these relations. a) R 1 R 2? {(a, b) a is required to read or has read b} b) R 1 R 2 {(a, b) a is required to read and has read b} c) R 1 R 2 {(a, b) either a is required to read b but has not read it or a has read b but is not required to} Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 16 / 1

17 Exercise 31 Let A be the set of students at your school and B the set of books in the school library. Let R 1 and R 2 be the relations consisting of all ordered pairs (a, b), where student a is required to read book b in a course, and where student a has read book b, respectively. Describe the ordered pairs in each of these relations. d) R 1 R 2 {(a, b) a is required to read b but has not read it} e) R 2 R 1? {(a, b) a has read b but is not required to} Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 17 / 1

18 Exercise 36 Relations 1 R 1 = {(a, b) R 2 a > b} 2 R 2 = {(a, b) R 2 a b} 3 R 3 = {(a, b) R 2 a < b} 4 R 4 = {(a, b) R 2 a b} 5 R 5 = {(a, b) R 2 a = b} 6 R 6 = {(a, b) R 2 a b} a) R 1 R 1 R 1 a c b(a > b b > c) b) R 1 R 2? R 1 a c b(a b b > c a > c) Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 18 / 1

19 Exercise 36-Cont... Relations 1 R 1 = {(a, b) R 2 a > b} 2 R 2 = {(a, b) R 2 a b} 3 R 3 = {(a, b) R 2 a < b} 4 R 4 = {(a, b) R 2 a b} 5 R 5 = {(a, b) R 2 a = b} 6 R 6 = {(a, b) R 2 a b} c) R 1 R 3 R 2 a c b(a < b b > c) d) R 1 R 4? R 2 a c b(a b b > c) Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 19 / 1

20 Exercise 36-Cont... Relations 1 R 1 = {(a, b) R 2 a > b} 2 R 2 = {(a, b) R 2 a b} 3 R 3 = {(a, b) R 2 a < b} 4 R 4 = {(a, b) R 2 a b} 5 R 5 = {(a, b) R 2 a = b} 6 R 6 = {(a, b) R 2 a b} e) R 1 R 5 R 1 a c b(a = b b > c a > c) f) R 1 R 6? R 1 a c b(a b b > c) Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 20 / 1

21 Exercise 36-Cont... Relations 1 R 1 = {(a, b) R 2 a > b} 2 R 2 = {(a, b) R 2 a b} 3 R 3 = {(a, b) R 2 a < b} 4 R 4 = {(a, b) R 2 a b} 5 R 5 = {(a, b) R 2 a = b} 6 R 6 = {(a, b) R 2 a b} g) R 2 R 3 R 2 a c b(a < b b c) h) R 3 R 3? R 3 a c b(a < b b < c) Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 21 / 1

22 Exercise 44 Which of the 16 relations on {0, 1}, which you listed in Exercise 42, are Relations {(0, 0)} 3. {(0, 1)} 4. {(1, 0)} 5. {(1, 1)} 6. {(0, 0), (0, 1)} 7. {(0, 0), (1, 0)} 8. {(0, 0), (1, 1)} 9. {(0, 1), (1, 0)} 10. {(0, 1), (1, 1)} 11. {(1, 0), (1, 1)} 12. {(0, 0), (0, 1), (1, 0)} 13. {(0, 0), (0, 1), (1, 1)} 14. {(0, 0), (1, 0), (1, 1)} 15. {(0, 1), (1, 0), (1, 1)} 16. {(0, 0), (0, 1), (1, 0), (1, 1)} a) reflexive? 8,13,14,16 b) irreflexive? 1,3,4,9 c) symmetric? 1, 2, 5, 8, 9, 12, 15, 16 d) antisymmetric? 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14 Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 22 / 1

23 Exercise 44-Cont... Which of the 16 relations on {0, 1}, which you listed in Exercise 42, are Relations {(0, 0)} 3. {(0, 1)} 4. {(1, 0)} 5. {(1, 1)} 6. {(0, 0), (0, 1)} 7. {(0, 0), (1, 0)} 8. {(0, 0), (1, 1)} 9. {(0, 1), (1, 0)} 10. {(0, 1), (1, 1)} 11. {(1, 0), (1, 1)} 12. {(0, 0), (0, 1), (1, 0)} 13. {(0, 0), (0, 1), (1, 1)} 14. {(0, 0), (1, 0), (1, 1)} 15. {(0, 1), (1, 0), (1, 1)} 16. {(0, 0), (0, 1), (1, 0), (1, 1)} e) Asymmetric? 1, 3, 4 f) Transitive? 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16 Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 23 / 1

24 Exercise 46 Let S be a set with n elements and let a and b be distinct elements of S. How many relations R are there on S such that a) (a, b) R? 2 n2 1 b) (a, b) R? 2 n2 1 c) no ordered pair in R has a as its first element? 2 n(n 1) d) at least one ordered pair in R has a as its first element? 2 n2 2 n(n 1) e) no ordered pair in R has a as its first element or b as its second element? 2 (n 1)(n 1) f) at least one ordered pair in R either has a as its first element or has b as its second element? 2 n2 2 (n 1)(n 1) Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 24 / 1

25 Exercise 52 Show that the relation R on a set A is antisymmetric if and only if R R 1 is a subset of the diagonal relation = {(a, a) a A}. ) If R is antisymmetric and (x, y) R R 1, then (x, y) R 1 implies that (y, x) R. Therefore, (x, y) R and (y, x) R, and since R is antisymmetric, then x = y, and (x, y). So we have shown that R R 1. Conversely, suppose that R R 1. If (x, y) R and (y, x) R, then (x, y) R R 1, so that x = y, and therefore R is antisymmetric Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 25 / 1