一、映射：
1.概念：设X、Y是两个非空集合，如果存在一个法则 f ,使得对X中的每个元素 x,按法则 f ，在Y中有唯一的确定的元素y与只对应，那么称 f 为从X到Y的映射，记作：
f ：X →Y，
其中y称为元素x（在映射 f 下）的像，并记作 f（x），即
y=f(x),
而元素x称为元素y（在映射 f 下）的一个原像；集合X成为映射 f 的定义域，记作 Df ,即 Df = X ；X中所有元素的像所组成的集合成为映射 f 的值域，记作Rf 或 f(X),即
Rf= f(X)={ f(x) | x∈ X }
2.理解：
2.1 构成映射的三要素：对应法则 f （使对每个x∈ X，有唯一确定的 、值域、定义域。
2.2 对每个 x∈ X ，元素 x 的像是唯一的；而对于y ∈ Rf ，元素原像不一定是唯一的，即Rf 
Y， 不一定Rf = Y。
I. Mapping:
1. Concept: Let X and Y be two non-empty sets. If there is a rule f such that for each element x in X, according to the rule f, there is a uniquely determined element y in Y that corresponds to only one, then Call f a mapping from X to Y, and write it as:
F: X → Y,
Where y is called the image of element x (under map f) and is written as f (x), ie
y=f(x),
The element x is called an elementary image of the element y (under the mapping f); the set X becomes the domain of the mapping f, denoted as Df, that is, Df = X; the set of images of all elements in X becomes the mapping f Range, denoted as Rf or f (X), ie
Rf= f(X)={ f(x) | x∈ X }
2. Understand:
2.1 Three elements that make up a mapping: the correspondence rule f (so that for each x ∈ X, there is a uniquely determined y = f (x) corresponding to it), the range, and the domain.
2.2 For each x ∈ X, the image of element x is unique; for y ∈ Rf, the original image of element y is not necessarily unique, that is, Rf Y, not necessarily = Y.
特殊例子：若Rf =Y ，则称 f 为X到Y上的映射或者满射。
若对X中任意两个不同的元素x1≠x2, 他们的像f(x1)≠f(x2),则称 f 为X到Y上的单射。
若映射 f 即是单射又是满射，则称 f 为一一映射（或双射）。
Special example: If Rf = Y, then f is called a mapping or surjective from X to Y.
If for any two different elements in X x1 ≠ x2, their image f (x1) ≠ f (x2), then f is called injective from X to Y.
If f is both injective and surjective, then f is called a one-to-one mapping (or bijective).
逆映射 
，定义域 Df -1 = Rf ，值域Rf -1= X 。
复合映射 f°g ：X → Z，( f°g )(x) = f[g (x) ],x∈ X.
Inverse mapping
，the domain D
f -1 = Rf ,
the range R
f -1= X .
Compound mapping f°g ：X → Z，( f°g )(x) = f[g (x) ],x∈ X.