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1、2.2 Set Operations1. Introduction(1)Definition 1 (page 121) Let A and B be sets. The union of the sets A and B, denoted by AB, is the set that contains those elements that either in A or in B, or in both. AB = x | xA / xB Example 1: 1,3,5 1,2,3 = 1,2,3,5Example 2:The union of the set of all computer
2、 science majors at your school and the set of all mathematics majors at your school is the set of students at your school who are majoring either in mathematics or computer science( or in both).离散数学杨争锋ch(2) Definition 2 (intersection) Let A and B be sets. The intersection of the sets A and B, denote
3、d by AB, is the set containing those elements in both A and B. AB = x | xA / xB 1,3,51,2,3 = 1,3离散数学杨争锋ch(3) Definition 3 (page 122) Two sets are called disjoint if their intersection is the empty set. Example 5 (page 87) Let A=1,3,5,7,9 and B=2,4,6,8,10. Since AB= , A and B are disjoint.离散数学杨争锋ch(5
4、) For two finite sets A and B, we have: |AB| = |A|+|B|-| AB | Please explain it via Venn Diagram.离散数学杨争锋ch(5) Definition 4 (the difference of two sets, 差集差集) Let A and B be sets. The difference of A and B, denoted by A-B, is the set that containing those elements that are in A but not in B. The diff
5、erence of A and B is also called the complement of B with respect to A (关于关于A的的集合集合B的补集的补集) A-B=x | xA / x notin B Please explain it via Venn Diagram. 离散数学杨争锋ch Example 6 (page 123) 1,3,5-1,2,3 = ? 1,2,3-1,3,5 = ? 离散数学杨争锋ch(6) Definition 5 (complement of a set, 补集补集) Let U be universal set (全集全集). T
6、he complement of the set A, denoted by A, is the complement of A with respect to U. A = U-A A = x | x notin A Please explain it via Venn Diagram.离散数学杨争锋ch Example 8 (page 124) A=a,e,i,o,u U-the set of letters of the English alphabet A = ? Example 9 (page 124) A-the set of positive integers greater t
7、han 10 U-all positive integers A = ?离散数学杨争锋ch2. Set identities (集合的恒等式集合的恒等式)(1)Introduction Table 1 (page 124) Note: We will explain the Chinese name (定律的中文定律的中文名字名字) of each law via blackboard.(2) Example 10 (page 125) Prove that (AB ) = A B.Solution: (a) Left Right (b) Right Left离散数学杨争锋ch(3) Use
8、set builder notation and logical equivalence to show that (AB ) = A BProof: See page 125离散数学杨争锋ch(4) Example 12 (page 125) Prove that A(BC) = (AB)(AC) for all sets A, B, C.Proof See book.离散数学杨争锋ch(4) Use a membership table to show that A(BC) = (AB)(AC)Proof See book (page 126).离散数学杨争锋ch(5)Let A, B,
9、and C be sets. Show that (A (B C) = (CB)AProof: By using the set identities proved previously. See book (page 126)离散数学杨争锋ch3. Generalized unions and intersections(1)Introduction The well-definedness of ABC and ABC Why? Reason: the associative law of and (AB)C = A(BC) A(BC) = (AB)C离散数学杨争锋ch(2) Exampl
10、e 15 Let A=0,2,4,6,8, B=0,1,2,3,4, and C=0,3,6,9. What are ABC and ABC?Solution: ABC = ? ABC =?离散数学杨争锋ch(3) Definition 6 The union of a collection of sets is the set that contains those elements that are members of at least one set in the collection. The notation: A1A2An = i=1n Ai离散数学杨争锋ch(4) Defini
11、tion 7 The intersection of a collection of sets is the set that contains those elements that are members of all the sets in the collection. The notation: A1 A2 An = i=1n Ai离散数学杨争锋ch(5) Let Ai=i, i+1, i+2,. What are i=1n Ai and i=1n Ai?Solution: See blackboard.Example 17(page 128)离散数学杨争锋chSummary (1)
12、Ways of proving set identities:(1)Using natural language to express the proof. The key point is: (a) Left Right (b) Right Left(2) Using set builder notation and logical equivalences to show the proof.离散数学杨争锋chSummary (2)(3) Using membership table to show the proof(4) Using those already achieved laws to show to proof离散数学杨争锋chExercisesSixth EditionPage 130: 4、12、16（a）、18（a）（e）、20、24、30、35、36、46Fifth Edition P94 ：4、8、12（a）、14（a）（e）、16、18、22、27、28、38离散数学杨争锋ch
