用户: Endeavour/抽象代数/Prerequisites and Preliminaries

1Sets, functions and relations

Definition 1.1. A class, membership or equality is a collection $A$ of objects such that given any object $x$ it is possible to determine whether or not $x$ is a member "or element of $A$".

Definition 1.2. A class $A$ is defined to be a set iff there exists a class $B$ such that $A\in B$.

Definition 1.3. A proper class is a class that is not a set. (e.g. the set of all sets is a proper class)

A class $A$ is a subclass of a class $B$ (written as $A\subset B$) provided:$$forallx\in A,x\in A⟹x\in B.$$

2The NBG axiomatic system

Axiom of extensionality: Two classes with the same elements are equal.
Axiom of class formation: For any statement $P(y)$ in the first-order predicate calculus involving a variable $y$, there exists a class $A$ such that $x\in A$ iff $x$ is a set and the statement $P(x)$ is true.
Axiom of empty set: There exists a set $\varnothing$ such that given any $x$, $x\notin \varnothing$.
Power axiom: 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 denoted $2^A$.
Basic concepts: Family of sets, union and intersection, disjoint, complement, DeMorgan’s Laws.

3Functions

A diagram of functionsis said to be commutative if $gf=h$, similarly the diagramis commutative if $kh=gf$.

Definition 3.1. Let $f:A\to B$ be a function, if $S\subset A$, the image of $S$ under $f$(denoted $f(S)$) is the class $\left\{b\in B|b=f(a)forsomea\in b\right\}$

Definition 3.2. A function $f:A\to B$ is said to be injective provided$$foralla,a^{\prime }\in A⟹f(a)\ne f(a^{\prime }).$$A function $f:A\to B$ is said to be surjective provided$$foreachb\in B,b=f(a)forsomeA.$$A function $f:A\to B$ is said to be bijective if it is both injective and surjective.

4Relations and Partitions

5Products

Definition 5.1. Let $\{A_i|i\in I\}$ be a family of sets indexed by a nonempty set $I$, the Cartesian product of the sets $A_i$ is the set of all functions $f:I\to {\bigcup }_{i\in I}{A}_i$ such that $f(i)\in A_i$ for all $i\in I$. Denoted ${\prod }_{i\in I}{A}_i$.

Theorem 5.2 (Universal property of products).