injective, surjective,bijective区别

文章源自https://www.mathsisfun.com/sets/injective-surjective-bijective.html

1 函数的定义

A function is a way of matching the members of a set "A" to a set "B":
A集合中的元素，匹配B集合中元素的方法叫函数

2 几种匹配方法

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我们仔细看看这几种模式

2.1 General Function 普通函数

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A General Function points from each member of "A" to a member of "B".

It never has one "A" pointing to more than one "B", so one-to-many is not OK in a function (so something like "f(x) = 7 or 9" is not allowed)

But more than one "A" can point to the same "B" (many-to-one is OK)

每个自变量都有值与之对应，多个自变量可以对应一个值，即多对一

2.2 Injective 单射

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A function f is injective if and only if whenever f(x) = f(y), x = y.

Injective means we won't have two or more "A"s pointing to the same "B".

So many-to-one is NOT OK (which is OK for a general function).

As it is also a function one-to-many is not OK

But we can have a "B" without a matching "A"

Injective is also called "One-to-One"

是单射，就是说不能出现多对一的情况，必须一对一，允许有值没有自变量对应。

2.3 Surjective 满射

A function f (from set A to B) is surjective if and only if for every y in B, there is at least one x in A such that f(x) = y, in other words f is surjective if and only if f(A) = B.

Surjective means that every "B" has at least one matching "A" (maybe more than one).

There won't be a "B" left out.

是满射，允许多对一，B中必须每一个值都有自变量

2.4 Bijective 双射

A function f (from set A to B) is bijective if, for every y in B, there is exactly one x in A such that f(x) = y

Bijective means both Injective and Surjective together.

Think of it as a "perfect pairing" between the sets: every one has a partner and no one is left out.

So there is a perfect "one-to-one correspondence" between the members of the sets.

(But don't get that confused with the term "One-to-One" used to mean injective).

是一一对应，有逆的存在