CNLCE: Sets

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.(http://en.wikipedia.org/wiki/Set_%28mathematics%29)

considered(adj)考慮過得
ubiquitous(adj)普遍存在的
derive(v) 取得,得到,衍生
Venn diagrams(n)范氏圖

Set theory begins with a fundamental binary relation between an object

o

and a set

A

. If

o

is a member (or element) of

A

, we write

o \in A

. Since sets are objects, the membership relation can relate sets as well.
A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set

A

are also members of set

B

, then

A

is a subset of

B

, denoted

A \subseteq B

. For example,

{1,2}

is a subset of

{1,2,3}

, but

{1,4}

is not. From this definition, it is clear that a set is a subset of itself; in cases where one wishes to avoid this, the term proper subset is defined to exclude this possibility.
Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The:

inclusion(n) 包括
subset(n)子集合
denote(v)表示
proper subset(n)真子集
arithmetic(adj)算術的 (n)算術

Union of the sets $A$ and $B$ , denoted $A \cup B$ , is the set of all objects that are a member of $A$ , or $B$ , or both. The union of ${1, 2, 3}$ and ${2, 3, 4}$ is the set ${1, 2, 3, 4}$ .

union 聯集

Intersection of the sets $A$ and $B$ , denoted $A \cap B$ , is the set of all objects that are members of both $A$ and $B$ . The intersection of ${1, 2, 3}$ and ${2, 3, 4}$ is the set ${2, 3}$ .

intersection 交集

Set difference of $U$ and $A$ , denoted $U \ A$ is the set of all members of $U$ that are not members of $A$ . The set difference ${1,2,3} \ {2,3,4}$ is ${1}$ , while, conversely, the set difference ${2,3,4} \ {1,2,3}$ is ${4}$ . When $A$ is a subset of $U$ , the set difference $U \ A$ is also called the complement of $A$ in $U$ . In this case, if the choice of $U$ is clear from the context, the notation $A c$ is sometimes used instead of $U \ A$ , particularly if $U$ is a universal set as in the study of Venn diagrams.

complement(n)(數學)補數
set difference 差集
U\A = U - A = A*

Symmetric difference of sets $A$ and $B$ is the set of all objects that are a member of exactly one of $A$ and $B$ (elements which are in one of the sets, but not in both). For instance, for the sets ${1,2,3}$ and ${2,3,4}$ , the symmetric difference set is ${1,4}$ . It is the set difference of the union and the intersection, $(A ∪ B) \ (A ∩ B)$ .

Symmetric difference 對稱差 (數位邏輯 :互斥或)

Cartesian product of $A$ and $B$ , denoted $A \times B$ , is the set whose members are all possible ordered pairs $(a, b)$ where $a$ is a member of $A$ and $b$ is a member of $B$ .

Cartesian product 笛卡兒積

Power set of a set $A$ is the set whose members are all possible subsets of $A$ . For example, the powerset of ${1, 2}$ is ${ {}, {1}, {2}, {1,2} }$ .

power set 幕集 (所有子集之集合)

Some basic sets of central importance are the empty set (the unique set containing no elements), the set of natural numbers, and the set of real numbers.

empty set 空集合

(http://en.wikipedia.org/wiki/Set_theory)

CNLCE

2011年5月22日星期日

Sets

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2011年5月22日 星期日

Sets

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張貼留言

2011年5月22日星期日