2. Sets and classes
Gödel-Bernays Set Theory (冯・诺伊曼-博内斯-歌德尔集合论)
class 类
A class is a collection of objects (elements) such that given any object \(x\) it is possible to determine whether or not \(x\) is a member (or element) of the class.
对于一个类，给定任意元素，可以确定该元素是否为这个类的成员。
belong 属于
\(x \in A\) means that \(x\) is an element of \(A\).
\(x\) 是 \(A\) 的一个元素。
not belong 不属于
\(x \not \in A\) means that \(x\) is not an element of \(A\).
\(x\) 不是 \(A\) 的一个元素。
equality 类等价的性质
- \(A = A\)
- \(A = B \Rightarrow B = A\)
- \(A = B\) and \(B = C \Rightarrow A = C\)
- \(A = B\) and \(x \in A \Rightarrow x \in B\)
axiom of extensionality 外延公理
Two classes with the same elements are equal.Formally, \([ x \in A \Leftrightarrow x \in B ] \Rightarrow A = B\)
set 集合
A class \(A\) is defined to be a set if and only if there exists a class \(B\) such that \(A \in B\).
proper class 真类
A class that is not a set is called proper class.
真类是不为集合的类。
axiom of class formation 分类公理
For any statement \(P(y)\) in the first-order predicate calculus involving a variable \(y\), there exists a class \(A\) such that \(x \in A\) if and only if \(x\) is a set and the statement \(P(x)\) is true. We denote this class \(A\) by \(\{ x | P(x) \}\), and refer to "the class of all \(x\) such that \(P(x)\)".
A example of proper class
Consider the class \(M = \{ X | X\ is\ a\ set\ and\ X \not \in X \}\)
For if \(M\) were a set, then either \(M \in M\) or \(M \not \in M\).
But by the definition of \(M\), \(M \in M\) implies \(M \not \in M\) and \(M \not \in M\) implies \(M \in M\).
Thus in either case the assumption that \(M\) is a set leads to an untenable paradox: \(M \in M\) and \(M \not \in M\)
subclass 子类
\(A\) is a subclass of class \(B\) (written \(A \subset B\)) provided:
By the axioms of extensionality and the properties of equality:
subset 子集
A subclass \(A\) of a class \(B\) that is itself a set is called a subset of \(B\).
A subclass of a set is a subset.
作为一个子类的集合被称为子集，集合的子集仍是子集。
empty set 空集
empty set is also written as null set, \(\emptyset\)
It's the set with no elements.
Given any \(x\), \(x \not \in \emptyset\).
The statement \(x \in \emptyset\) is always false.
The implication \(\eqref{ref1}\) is always true when \(A = \emptyset\).
\(\emptyset \subset B\) for every class \(B\)
空集是没有任何元素的集合
proper subclass 真子类
\(A\) said to be a proper subclass of \(B\) if \(A \subset B\) but \(A \not = \emptyset\) and \(A \not = B\)
与之前所学真子集的对比
之前所学的真子集不要求\(A \not = \emptyset\)
power axiom and power set 幂集公理与幂集
For every set \(A\), the class \(P(A)\) of all subsets of \(A\) is itself a set.
\(P(A)\) is called the power set of \(A\), it is also denoted \(2^A\).
family of sets 集族
A family of sets indexed by (the nonempty class) \(I\) is a collection of sets \(A_i\), one for each \(i \in I\) (denoted \(\{A_i | i \in I\}\))
union and intersection of a family 集族的并与交
Give such a family, it's union and intersection are defined to be respectively the classes
If \(I\) is a set, then suitable axioms insure that \(\bigcup_{i \in I} A_i\) and \(\bigcap_{i \in I} A_i\) are actually sets.
disjoint 不交集
If \(A \cap B = \emptyset\), \(A\) and \(B\) are said to be disjoint.
relative complement 相对补
If \(A\) and \(B\) are classes, the relative complement of \(A\) in \(B\) is the following subclass of \(B\):
complement 补
If all the classes under discussion are subsets of some fixed set \(U\) (called the universe of discussion), then \(U - A\) is denoted \(A'\) and called simply the complement of \(A\).
statements 一些结论
\(\eqref{ref3}\) are DeMorgan's Laws