We deal with different objects in math such as numbers, functions(函数), operators(算子), and so on. It is important to indicate what kinds of objects we are dealing with before starting our study. Therefore, set plays a fundamental role in math. With different sets, we might have very different results for the same problem. For example, a quadratic equation(二次方程) ax^2+bx+c=0 with a \neq 0 and b^2-4ac<0 has no root(根) in \mathbb{R}(实数) , but two different roots in \mathbb{C}(虚数).

Here we only discuss naive set theory, which is presumed and used in most branches of mathematics. There is a much more strict set theory, which involves a lot of mathematical logic. Because of the reason we mentioned above, it has become the foundation of every other part of mathematics.

Set is one of basis conceptions(概念), which other definitions (定义) depend on. There is no strict (mathematical) way to define what is called a set. It is as if we have to define 1+1=2 and some other laws to define addition(加法). One could also choose to define 1+2=3 firstly, but anyway, there needs to be a initial conception.  Roughly speaking (大概地说), a set is a collection of objects (一些目标对象在一起的整体/汇合) we are interested in, which has specific properties.

1. Terminology and Notation 术语和记号

We call parts of a set elements (元素). We normally use letters in upper case (大写字母) to represent sets (collection) and letters in lower case (小写字母) to represent elements. To say a is a part of A, the following ways can be used:

  • a is a member of A;
  • a is an element of A;
  • a is contained in A;
  • a is included in A;
  • A contains a;
  • A includes a.

We write it as a \in A or A \ni a. If a is not a member of A, we can write it as a \notin A. For example, a \in \{a,b,c\}, d \notin \{a,b,c\}.

We can describe a set in several ways:

  • Specifying members. For example A=\{a,b,c\}.
  • Characterizing (描述元素特征) members. For example A=\{x|x\ \text{is a real number}\}, B=\{x|x \leq 1\}.

In a set, we ignore repetition (重复)and arrangement (顺序). For example, \{a, b,c\}=\{c,a,b,a,b,c\}. When we count the number of elements of a set, no matter how many times an element appears, it is only counted once. Therefore, the above set has 3 elements.

An empty set (空集)is a set which contains no element. We use \emptyset to represent such a set.

Two sets, A\ \text{and}\ B, are said being the same if and only if each element of A is a member of B and each member of B is a member of AThis is the commonest way to prove that two sets are the same.

2. Set relationship and set operations 集合之间的关系以及集合运算

We say A is a subset of B (or B is a superset of A) if any member of A is also a member of B. We denote (表示)it by A \subset B, A \subseteq B, B \supset A, or B \supseteq A.

If, additionally, there is a member of B which is not a member of A, then we call A a proper subset of B (or B is a proper superset of A). We denote it by A \subsetneq B, A \subsetneqq B, B \supsetneq A, or B \supsetneqq A. For example, \{a,b\} \subset \{b,a\} \subsetneq \{a,c,b\}.

Remark: \subset, \subsetneq, \supset, and \supsetneq are like <, \leq, > and \geq between real numbers, which make us be able to compare different sets. But not all pairs of sets can be compared in this sense. For example, A=\{a,b\}, B=\{b,c\}.  We will discuss more about this later in the ‘ordering’ section.

A power set (幂集)of A, denoted by \mathcal{P}(A), is the set of all subsets of A. For example, A=\{a,b,c\}, \mathcal{P}(A)=\{\emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}\}. It is obvious that the power set of a N elements set has 2^N elements.

Given two sets A\ \text{and}\ B, we call C the union (并集)of A\ \text{and}\ B, denoted by C=A \cup B, if C contains and only contains elements from either A or B, i.e. A \cup B=\{x|x \in A\ \text{or}\ x \in B\}.

We call D the intersection (交集)of A\ \text{and} B, denoted by D=A \cap B, if D contains and only contains elements from both A and B, i.e. A \cap B=\{x|x \in A\ \text{and}\ x \in B\}.

Given two sets A\ \text{and} B, we define the compliment of A with respect to B(A在B里的补集) as the set which contains all of the elements that are in B but not in A. We write it as B\backslash A or B-A, namely(即), B\backslash A=\{x|x \in B\} \cap \{x|x \notin A\}.

Remark: Since we have not indicated the whole space, \{x|x \notin A\} could be a tremendously huge set.

Since we already used letters in the upper case to represent sets, we choose Fraktur font to avoid confusion when we write a family (collection) of sets, i.e. a set of sets. For example, A, B\ \text{and} C are three sets. \mathfrak{A}=\{A,B,C\}. (\mathfrak{A} is A in the Fraktur font.) For example,

\bigcup \mathfrak{A}=\bigcup_{A \in \mathfrak{A}} A=\{x|x \in A\ \text{for some}\ A \in \mathfrak{A}\}.

Theorem 1: 

  • A \cap B=B \cap A; A \cup B=B \cup A.
  • (A \cap B) \cap C=A \cap (B \cap C).
  • (A \cup B) \cup C=A \cup (B \cup C).
  • A \cap (B \cup C)=(A \cap B) \cup (A \cap C).
  • A \cup (B \cap C)=(A \cup B) \cap (A \cup C).
  • A \backslash (B \cap C)=(A \backslash B) \cup (A \backslash C).
  • A \backslash (B \cup C)=(A \backslash B) \cap (A \backslash C).

The last two formulas are also called the De Morgan’s laws.

Reviewed by Rachel Reed and Karthick Alagarsamy.