sot 集合 1.2集合 Sets 121集合的基本概念 Concepts of Sets 2/24/202111:37PM Deren Chen Zhejiang univ
S e t s 集合 2/24/2021 11:37 PM Deren Chen, Zhejiang Univ. 1 1.2 集合 Sets 1.2.1 集合的基本概念 Concepts of Sets
DEFINITION 1 80t8 集合 The objects in a set are also called the elements or members of the set, a set is said to contain its elements Null Set: There are no anything in the set O={x|x∈P(x)∧~(x∈P(x)) 2/24/202111:37PM Deren Chen Zhejiang univ 2
S e t s 集合 2/24/2021 11:37 PM Deren Chen, Zhejiang Univ. 2 DEFINITION 1. The objects in a set are also called the elements, or members, of the set. A set is said to contain its elements. Null Set: There are no anything in the set. Ø={x | xP(x) ∧ (x P(x) )}
EXAMPLE1 80t8 集合 The set v of all vowels in the english alphabet can written as --- V=a, e, i, 0, u. 2/24/202111:37PM Deren Chen Zhejiang univ 3
S e t s 集合 2/24/2021 11:37 PM Deren Chen, Zhejiang Univ. 3 EXAMPLE 1 The set V of all vowels in the English alphabet can written as ---- V = {a, e, i, o, u}
EXAMPLE 2 80t8 集合 The set o of odd positive integers less than 10 can be xpressed by O={1,3,5,7,9} O={x|x>=1∧x<=10∧ x is odd number} A={a,b,c,…,n}枚举法 A={x|x∈P(x)}谓词公式法 2/24/202111:37PM Deren Chen Zhejiang univ
S e t s 集合 2/24/2021 11:37 PM Deren Chen, Zhejiang Univ. 4 EXAMPLE 2 The set O of odd positive integers less than 10 can be expressed by O = {1, 3, 5, 7, 9}. O = {x | x >=1 ∧ x <=10 ∧x is odd number}. A = {a, b, c, …… n} 枚举法 A = {x | xP(x)} 谓词公式法
DEFINTION 2. 80t8 集合 Two sets are equal if and only if they have the same elements. Presented as A=B otherwise a= B A=B分Vx(x∈A+x∈B) 2/24/202111:37PM Deren Chen Zhejiang univ
S e t s 集合 2/24/2021 11:37 PM Deren Chen, Zhejiang Univ. 5 DEFINTION 2. Two sets are equal if and only if they have the same elements. Presented as A=B otherwise A B A = B x ( x A x B)
EXAMPLE 5 80t8 集合 The sets ( 1, 3, 5 and 3, 5, 1 are equal, since they have the same elements note that the order in which the elements of a set are listed does not matter. note also that it does not matter if an element of a set is listed more than once, so that (1, 3, 3,, 5 is the same as the set (1, 3, 5 since they have the same elements. 2/24/202111:37PM Deren Chen Zhejiang univ
S e t s 集合 2/24/2021 11:37 PM Deren Chen, Zhejiang Univ. 6 EXAMPLE 5 The sets {1, 3, 5} and {3, 5, 1} are equal, since they have the same elements. Note that the order in which the elements of a set are listed does not matter. Note also that it does not matter if an element of a set is listed more than once, so that {1, 3, 3, 3, 5, 5, 5, 5} is the same as the set {1, 3, 5} since they have the same elements
EXAMPLE 5 80t8 集合 l、集合中的元素互异 2、集合中的元素无次序和大小之分 3、集合中的元素不一定同类 集合中的元素也可以是集合 2/24/202111:37PM Deren Chen Zhejiang univ
S e t s 集合 2/24/2021 11:37 PM Deren Chen, Zhejiang Univ. 7 EXAMPLE 5 1、集合中的元素互异 2、集合中的元素无次序和大小之分 3、集合中的元素不一定同类 4、集合中的元素也可以是集合
DEFINITION 3 80t8 集合 The set a is said to be a subset of B if and only if every element of a is also an element of B. we use the notation acb to indicate that a is a subset of the set B A是B的子集、 B包含A、 A包含在B中 2/24/202111:37PM Deren Chen Zhejiang univ
S e t s 集合 2/24/2021 11:37 PM Deren Chen, Zhejiang Univ. 8 DEFINITION 3. The set A is said to be a subset of B if and only if every element of A is also an element of B. We use the notation AB to indicate that A is a subset of the set B. A是B的子集、 B包含A、 A包含在B中
sot 集合 The set a is said to be a proper subset of B ifAcB and there is a∈ B and agA A是B的真子集、 B真包含A、 A真包含在B中 2/24/202111:37PM Deren Chen Zhejiang univ
S e t s 集合 2/24/2021 11:37 PM Deren Chen, Zhejiang Univ. 9 The set A is said to be a proper subset of B if AB and there is aB and aA A是B的真子集、 B真包含A、 A真包含在B中
sot 集合 定理1:A=B当且仅当AcB∧BcA 定理2:对任意的集合A,AgA 定理3:对任意的集合A、B、C 如果AcB,BcC,则AgC 定理4:对任意的集合A,⑦cA 定理5:空集是唯一的 2/24/202111:37PM Deren Chen Zhejiang univ
S e t s 集合 2/24/2021 11:37 PM Deren Chen, Zhejiang Univ. 10 定理1：A = B 当且仅当A B B A 定理2：对任意的集合A， A A 定理3：对任意的集合A、B、C， 如果 A B， B C，则 A C 定理4：对任意的集合A， A 定理5：空集是唯一的