每周一交作业,作业成绩占总成绩的 10% 平时不定期的进行小测验,占总成绩的 20% 期中考试成绩占总成绩的20%;期终考 试成绩占总成绩的50%
每周一交作业，作业成绩占总成绩的 10%； 平时不定期的进行小测验，占总成绩的 20%； 期中考试成绩占总成绩的20%；期终考 试成绩占总成绩的50%
A∪B=AUC≠B=C cancellation law×。 Example:A={1,2,3},B={3,4,5},C={4,5},B≠C, ButA∪B=AUC ={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 AeB 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. ifAOB=AOC. then B=c Distributive laws and De Morgan's laws: B∩(A1UA2U.UAn)=(B∩A1)U(B∩A2)U.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 = =
Chapter 2 Relations Definition 2.1: An order pair (a, b) is a listing of the objects a and b in a prescribed order, with a appearing first and b appearing second. Two order pairs(a, b)and (c, d) are equal if only if a=c and b=d {a,b}={b,a}, order pairs: (a, b)=(b, a)unless a=b aa
Chapter 2 Relations Definition 2.1: An order pair (a,b) is a listing of the objects a and b in a prescribed order, with a appearing first and b appearing second. Two order pairs (a,b) and (c, d) are equal if only if a=c and b=d. {a,b}={b,a}， order pairs: (a,b)(b,a) unless a=b. (a,a)
Definition 2.2: The ordered n-tuple (alayy.,a) is the ordered collection that has a, as its first element, a, as its second element. and a as its nth element. Two ordered n-tuples are equal is only if each corresponding pair of their elements ia equal, i.e. a.a 1929 gap)=(b1,b2,.,b) if only il is for i=1.2.n a =b. fo
Definition 2.2: The ordered n-tuple (a1 ,a2 ,…,an ) is the ordered collection that has a1 as its first element, a2 as its second element,…, and an as its nth element.Two ordered n-tuples are equal is only if each corresponding pair of their elements ia equal, i.e. (a1 ,a2 ,…,an )=(b1 ,b2 ,…,bn ) if only if ai=bi , for i=1,2,…,n
Definition 2.3: let a and b be two sets The Cartesian product of A and B, denoted by AXB, is the set of all ordered pairs(a,b) where a∈ A and b∈B. Hence A×B={(a,b)a∈ A and b∈B Example: Let A=(1, 2, B=x, y, C=a, b, c) A×B={(1,x),(1,y),(2,x)2,y)}; B×A={(x,1),(x,2),y,1),(y,2)}; B×A≠AXB commutative laws X
Definition 2.3: Let A and B be two sets. The Cartesian product of A and B, denoted by A×B, is the set of all ordered pairs ( a,b) where aAand bB. Hence A×B={(a, b)| aAand bB} Example: Let A={1,2}, B={x,y},C={a,b,c}. A×B={(1,x),(1,y),(2,x),(2,y)}; B×A={(x,1),(x,2),(y,1),(y,2)}; B×AA×B commutative laws ×
A×C={(1,a),(1,b),(1,c),(2,a)2(2,b),(2, c)} A×A={(1,1),(1,2),(2,1),(2,2)}。 A×=×A=8 Definition 2.4: LetA,A..a be sets. The Cartesian product of A142…An denoted A1×A2×…XAn, is the set of all ordered n-tuples(a1,a2,.ga,)where aEA: for i=1.2...n. Hence A1×A2×…×An={(a1a2,…,n)a∈A ,i=1,2,,n}
A×C={(1,a),(1,b),(1,c),(2,a),(2,b),(2, c)}; A×A={(1,1),(1,2),(2,1),(2,2)}。 A×=×A= Definition 2.4: Let A1 ,A2 ,…An be sets. The Cartesian product of A1 ,A2 ,…An , denoted by A1×A2×…×An , is the set of all ordered n-tuples (a1 ,a2 ,…,an ) where aiAi for i=1,2,…n. Hence A1×A2×…×An ={(a1 ,a2 ,…,an )|aiAi ,i=1,2,…,n}
Example:A×BC={(1,x,a),(1,x,b),(1,x,C)(1,y,a), (1y,b),(1,y,c),(2,x,a),(2,x,b),(2,x,c),(2,y,a),(2,y,b), (2y,c)} IfA= A for i=1,2,,n, then a1×A,×…× A, by an Example: Let A represent the set of all students at an university, and let B represent the set of all course at the university. What is the cartesian product of A×B? The Cartesian product of A XB consists of all the ordered pairs of the form(a, b), where a is a student at the university and b is a course offered at the university. The set A XB can be used to represent all possible enrollments of students in courses at the university
Example:A×B×C={(1,x,a),(1,x,b),(1,x,c),(1,y,a), (1,y,b), (1,y,c),(2,x,a),(2,x,b),(2,x,c),(2,y,a),(2,y,b), (2,y,c)}。 If Ai=A for i=1,2,…,n, then A1×A2×…×An by An . Example：Let A represent the set of all students at an university, and let B represent the set of all course at the university. What is the Cartesian product of A×B? The Cartesian product of A×B consists of all the ordered pairs of the form (a,b), where a is a student at the university and b is a course offered at the university. The set A×B can be used to represent all possible enrollments of students in courses at the university