1、Discrete Mathematics,Chapter 7 Relations (關係),大葉大學 資訊工程系 黃鈴玲,Ch8-2,7.1 Relations and their properties.,表示兩集合間元素的關係，最直覺的方式就是使用序對(ordered pair) (有順序的配對)。由序對構成的集合稱為二元關係(binary relation)。,Example 1.A : the set of students in your school.B : the set of courses.R = (a, b) : aA, bB, 學生a 選修了課程 b ,Def 1Let A
2、 and B be sets. A binary relation from A to B is a subset R of AB = (a, b) : aA, bB .,Ch8-3,Example 3. Let A=0, 1, 2 and B=a, b, then R = (0,a),(0,b),(1,a),(2,b) is a relation from A to B.,用圖形來表示關係：,Ch8-4,Example: A : 男生, B : 女生, R : 夫妻關系 A : 城市, B : 州或省 R : 屬於 (Example 2),Note. Relations vs. Functi
3、onsA relation can be used to express a 1-to-manyrelationship between the elements of the sets A and B.(Function 不可一對多，只可多對一),Def 2. A relation on the set A is a subset of A A (i.e., a relation from A to A).,Ch8-5,Example 4. Let A be the set 1, 2, 3, 4. 則 R = (a, b)| a divides b 裡面包含哪些序對?,Sol :,R = (
4、1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4) ,Ch8-6,Example 5. 考慮下列定義在Z上的關係.,R1 = (a, b) | a b R2 = (a, b) | a b R3 = (a, b) | a = b or a = -b R4 = (a, b) | a = b R5 = (a, b) | a = b+1 R6 = (a, b) | a + b 3 ,哪些關係包含了序對(1,1), (1,2), (2,1), (1,-1),及 (2,2)?,Sol :,Exercise 7.1,2. 列出在集合1, 2, 3, 4
5、, 5, 6上 關係 R=(a,b) | a整除b 中所有的有序數對。,Ch8-7,1. 列出由A=0,1, 2, 3, 4到 B=0, 1, 2, 3關係中所有 的有序數對，其中關係 R 定義如下： (a) a = b (b) a + b = 4 (e) gcd(a, b)=1,Ch8-8,Def 3. A relation R on a set A is called reflexive (反身性) if (a,a)R for every aA.,Example 7. 考慮下列定義在 1, 2, 3, 4 上的關係: R2 = (1,1), (1,2), (2,1) R3 = (1,1),
6、 (1,2), (1,4), (2,1), (2,2), (3,3), (4,1), (4,4) R4 = (2,1), (3,1), (3,2), (4,1), (4,2), (4,3) 哪些關係具備反身性(reflexive)？,Sol : (1, 1), (2, 2), (3, 3), (4, 4)都必須屬於R, 關係的性質：, R3,Ch8-9,Example 8. 下列定義在Z上的關係，哪些具備反身性(reflexive)？R1 = (a, b) | a b R2 = (a, b) | a b R3 = (a, b) | a = b or a = -b R4 = (a, b) | a
7、 = b R5 = (a, b) | a = b+1 R6 = (a, b) | a + b 3 Sol : 所有Z中的元素a，(a,a)都要屬於R，R才有反身性 (0,0)R2, R1, R3 and R4,(0,0)R5,(2,2)R6,Ch8-10,Def 4. (1) A relation R on a set A is called symmetric (對稱) if for a, bA, (a, b)R (b, a)R.(2) A relation R on a set A is called antisymmetric (反對稱) if for a, bA, (a, b)R an
8、d (b, a)R a = b.,即若 ab且(a,b)R (b, a)R,Ch8-11,Example 10. 下列關係，哪些有對稱性(symmetric)或反對稱性(antisymmetric)? R2 = (1,1), (1,2), (2,1) R3 = (1,1), (1,2), (1,4), (2,1), (2,2), (3,3), (4,1), (4,4) R4 = (2,1), (3,1), (3,2), (4,1), (4,2), (4,3) Sol : 對稱：若有序對(a,b)，就要有序對(b,a) 反對稱：若有序對(a,b)且ab，就不能有(b,a)R2, R3 are s
9、ymmetricR4 are antisymmetric.,Ch8-12,Def 5. A relation R on a set A is called transitive(遞移) if for a, b, c A, (a, b)R and (b, c)R (a, c)R.,Ch8-13,Example 13. 下列關係有哪些具備遞移性(transitive)? R2 = (1,1), (1,2), (2,1) R3 = (1,1), (1,2), (1,4), (2,1), (2,2), (3,3), (4,1), (4,4) R4 = (2,1), (3,1), (3,2), (4,1
10、), (4,2), (4,3) Sol : 檢查：若(a, b)R 且(b, c)R ，則 (a, c)也必須R R2 沒有遞移性，因 (2,1) R2 and (1,2) R2 but (2,2) R2. R3 沒有遞移性，因 (2,1) R3 and (1,4) R3 but (2,4) R3. R4 is transitive.,Exercise 7.1,4.對下列定義於所有人形成集合上的關係，判斷 是否具有反身性、對稱性、反對稱性和遞移性。 當 (a,b)R 若且唯若 (a) a 比 b 高 (b) a 與 b 生於同一天 (c) a 與 b 的名字相同 (d) a 與 b 有相同的祖
11、父母,Ch8-14,Exercise 7.1,7. 對下列定義於所有整數集合上的關係，判斷是否 具有反身性、對稱性、反對稱性和遞移性。 (a) R=(x, y) | x y, where x, yZ (b) R=(x, y) | xy 1, where x, yZ (c) R=(x, y) | x = y + 1 or x = y - 1, where x, yZ (d) R=(x, y) | x y (mod 7) , where x, yZ ,Ch8-15,Ch8-16,Example 17. Let A = 1, 2, 3 and B = 1, 2, 3, 4.The relation
12、R1 = (1,1), (2,2), (3,3) and R2 = (1,1), (1,2), (1,3), (1,4) can be combined to obtain R1 R2 R1 R2 = (1,1) R1 R2 = (2,2), (3,3) R2 R1 = (1,2), (1,3), (1,4) R1 R2 = (2,2), (3,3), (1,2), (1,3), (1,4),對稱差(symmetric difference), 即 (A B) (A B),Ch8-17,補充 : antisymmetric 跟 symmetric可並存,只要R中沒有(a, b)且ab即可,例：令A = 1,2,3, 給出一個定義在A上的關係R，R需同時具備對稱性、反對稱性，但不具備反身性。Sol : R = (1,1), (2,2),
