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injective, surjective,bijective区

injective, surjective,bijective区

作者: readilen | 来源:发表于2020-05-24 18:04 被阅读0次

    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

    image.png

    我们仔细看看这几种模式

    2.1 General Function 普通函数

    image.png
    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

    image.png

    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

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