每周三交作业,作业成绩占总成绩的15%; 平时不定期的进行小测验,占总成绩的 15%; 期中考试成绩占总成绩的20%;期终考试成绩 占总成绩的50% zhymafudan. edu.cn 张宓1321201027@udanedu.cn BBS id: abchjsa bc软件楼1039 杨侃10302010007@fudan.edu.cn liy(fudan, edu.cn李弋
每周三交作业，作业成绩占总成绩的15%； 平时不定期的进行小测验，占总成绩的 15%； 期中考试成绩占总成绩的20%；期终考试成绩 占总成绩的50% zhym@fudan.edu.cn 张宓 13212010027@fudan.edu.cn BBS id:abchjsabc 软件楼1039 杨侃 10302010007@fudan.edu.cn liy@fudan.edu.cn 李弋
A∪B=AUC≠B=C cancellation law×。 Example:A={1,2,3},B={3,4,5},C={4,5},B≠C, ButA∪B=A∪C={1,2,3,4,5} Example:A={1,2,3},B={3,4,5},C={3},B≠C, ButA∩B=A∩C={3} A-B=A-C=B=C cancellation law symmetric difference
A∪B=A∪C ⇏ B=C cancellation law 。 Example:A={1,2,3},B={3,4,5},C={4,5}, BC, But A∪B=A∪C={1,2,3,4,5} Example: A={1,2,3},B={3,4,5},C={3},BC, But A∩B=A∩C={3} A-B=A-C ⇏B=C cancellation law :symmetric difference
The symmetric difference of A and b, write AOB is the set of all elements that are in A or B, but are not in both A and B, i.e. AGB=(A∪B)(A∩B)。 (A∪B)-(A∩B)=(A-B)U(B-A) A⊕B
The symmetric difference of A and B, write AB, is the set of all elements that are in A or B, but are not in both A and B, i.e. AB=(A∪B)-(A∩B) 。 (A∪B)-(A∩B)=(A-B)∪(B-A)
Pr oof:Left=(4∪B)-(4∩B)=(AUB)∩(A∩B) =(A∪B∩(A∪B)( De morgan'slms) =(∪B)∩A)∪(AUB)∩B)( distributi ve laws) =(A∩A)U(B∩A)∪(A∩B)∪(B∩B)( distributi ve laws) =(U(B-A)∪(A-B)∞)( complement laws) (A-B)U(B-A(identical laws, commutativ e laws
Pr oof : Left = (A B) − (A B) = (A B)(A B) = (A B)(A B) (De Morgan's laws) = ((A B) A)((A B) B) (distributi ve laws) = ((A A)(B A))(A B)(B B) (distributi ve laws) = ( (B − A))((A− B)) (complement laws) = (A− B)(B − A) (identical laws,commutativ e laws)
Theorem 1. 4: if AOB=AeC. then B=c Distributive laws and De Morgan's laws B∩(A1UA2UUA)=(B∩A1)U(BnA2)U…∪(B∩A B∪(A1nA2n.∩A)=(BUA1)∩GB∪A2)n…n(B∪An) ∩4=∪4 ∪4=∩4 i=1
Theorem 1.4: if AB=AC, then B=C Distributive laws and De Morgan’s laws: B∩(A1∪A2∪…∪An)=(B∩A1)∪(B∩A2)∪…∪(B∩An) B∪(A1∩A2∩…∩An)=(B∪A1)∩(B∪A2)∩…∩(B∪An) n i i n i i n i i n i Ai A A A =1 =1 =1 =1 = =