Set Theory Sets 許多數學上的討論中（例如： algebra 、 analysis 、 geometric 等等） 經常藉助集合論中的符號或術語來說 明。集合論是十九世紀後期由 Boole ( 布爾； 1815~1864) 及 Cantor ( 康托爾； 1845 ~ 1918) 所發展出來的。

Set Theory Boole ( 布爾； 1815~1864) 英國數學家及邏輯學家，以形式邏輯方 面的開創性工作最為著名。對分析學、微 分方程、代數和機率論等方面貢獻良多。 1854 年出版 “An Investigation of the Lows of Thought” 為符號邏輯（ symbolic logic ）方 面第一部可行的有系統的著作。

Set Theory Cantor( 康托爾； 1845~1918) 生於俄國聖彼得堡，但在德國長大。 以集合論的創始人，和對古典分析與拓 樸學的基本性貢獻著名。他及其門徒於 1874 至 1885 年間奠定了現代集合論的基 礎。創立了實數等效於有理數的科西序 列（ Cauchy Sequence ）之類的定義，開 集和閉集的定義以及超越數（ transfinite numbers ）的理論。由於生前不受重視， 患憂鬱症死於精神療養院。

Set Theory Definition of a Set A (undefined) term set is a well-defined collection of objects.

Set Theory Ex1.

Set Theory Definitions: The object of a set S is called an element of S. |S| = the number of elements in a set S. If |S| is finite, then we say this set S is a finite set. Otherwise, S is an infinite set. A set contains no element is called the empty set, denoted by Ø. Two sets A and B are equal, denoted by A = B, if and only if they contain exactly the same elements.

Set Theory Ex2. A = {1, 2, 3}, C is the set of all nonnegative integers B = {0, 1, 2, 3, …} = {x |x is a nonnegative integers.} (set-builder notation) Then

Set Theory Ex3. Let X = { {1}, {2, 3}, 1, 2 }. Then (1) |X | = (2)

Set Theory Definitions: Let A and B be two sets. We say A is a subset of B if and only if x  B,  x  A. Denoted by A  B. A proper subset of S is a subset of S which is neither S nor Ø. A power set,  (A)={ X | X  A }, of a set A is the set of all subsets of A. We may assume that all the sets we are dealing with are subsets of some universal set, denoted by U.

Set Theory Ex4. Let A = {1, 2, 3}, B = {0, 1, 2, 3, …}. Then (1) (2)  (A)=

Set Theory Note: Let A and B be two sets. A = B if and only if A  B and B  A. S  S, for any set S. | Ø | = 0. |  (A)| > 0, since Ø   (A) for any set A. |  (A)|= 2 |A| if A is a finite set. The empty set is a subset of any set.

Set Theory Operations of sets Let A and B be two subsets of the universal set U. The union of A and B is the set A  B = {x | x  A or x  B }. The intersection of A and B is the set A  B = {x | x  A and x  B }. the difference of B in A is the set A – B = {x | x  A and x  B }. the complement of A is the set A’ = U – A = {x | x  U and x  A }.

Set Theory Ex5. Let A = {2, 4, 6} and B = {4, 5, 6, 7}. Then A  B = A  B = A – B = B – A =

Set Theory Ex6. Let U be the set of integers. A = {x | x is an even integer.} and B = {x | x is a positive integer.} Then B – A = A – B = A’ = B’ =

Set Theory Note Commutative property A  B = B  A; A  B = B  A Associative property A  (B  C) = (A  B)  C; A  (B  C) = (A  B)  C Distributive property A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C) De Morgan’s Laws (A  B)’ = A’  B’; (A  B)’ = A’  B’

Set Theory Venn Diagram A  B A  B

Set Theory A – B A’

Set Theory A  B = Ø A  B

Set Theory Notations Z + : the set of all positive integers Z : the set of all integers Q : the set of all rational numbers R : the set of all real numbers C : the set of all complex numbers

Set Theory The Relation of Number Sets

Set Theory Ex7. Show that A  (B  C) = (A  B)  (A  C).