Discrete mathematics single shot, full shot and double shot

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Contents of this article

• 1、 What is mapping ？
• 2、 Classification of mappings
  • 2.1 Single shot
  • 2.2 Full shot
  • 2.3 Double shot
  • 2.4 It is neither simple nor surjective , But mapping
• 3、 Have you got it ？
• 4、 Experience

1、 What is mapping ？

We consider this relationship ： about aggregate X Every element in , There are The only one belongs to the set Y The elements in Be pointed at by it , We call this relationship mapping （ Britain ：mapping, Japan ： mapping （しゃぞう））. This is a very popular language to explain the definition of mapping , And I believe we are all required in high school mathematics 1 I learned it inside , The concept of mapping must be familiar to us ！ From this definition , Can you get To what information ？ ①“X In the collection every last Elements ”： If there are collections X The element of does not correspond to the set Y Of an element of , Is not a mapping . ②“ There are Unique Y With the corresponding ”： If there are collections X The element of also points to the collection Y Of two or more elements in , Is not a mapping .

therefore , To get into today's topic —— Single shot 、 The distinction between surjection and bijection , First you have to judge , Is it a collection X All elements in point to , And only points to the set Y One of the elements ？ Only if（ The result of the judgment ==true） when , Only then can we make the next step of distinguishing and judging ;else, It's not even a mapping , What else to judge ？ This is like comparison. , If you are a senior three student , You don't even belong to grade one , I have to ask if you are a senior one （7） Ben ？

Summary ： If the pointing relationship between the elements of two sets satisfies the following two conditions , It is a mapping relationship . Only then can we further judge , What kind of mapping is it . (1) aggregate X All elements in point to a collection Y Some element in , And no one does not point to ; (2) aggregate X All elements in point only to collections Y An element in , And no one points to two or more .

2、 Classification of mappings

2.1 Single shot

All being X The elements in point to Y Elements in , All are Only by one X The element of the Of , Not by two or more X The element of the , Then this mapping relationship is called Single shot （ Britain ：injection, Japan ： Shooting （たんしゃ））. Single shot does not consider whether or not Y All the elements in it are X The element of points to . There can be Y The element of is not defined by any X The element of the , It doesn't affect that it is a single shot .

for instance .

For example, watching movies . Yes “ The audience ”（X） and “ seat ”（Y） Two sets . Now we know that , Every audience must sit in one seat , And he only sits in one seat , So this correspondence is first of all a mapping . And because each seat can only be matched by one audience , It is impossible for two people to sit in one seat , And not all seats must have corresponding audience , There may also be vacant seats , So this correspondence is a kind of Single shot Relationship .

Audience and cinema seats （ Single shot ）： ① Or no one is sitting ; ② Or just one person sitting . （ Or not be pointed at by any audience , Or be pointed at by only one audience ）

2.2 Full shot

Y In the assembly All elements are X The element in the points to , aggregate Y There is no element in that is not collected X The element of the , Then this mapping relationship is called Full shot （ Britain ：surjection, Japan ： All shot （ぜんしゃ））. The surjection does not consider pointing to the same Y How many X. As long as all Y Are pointed to , Is a full shot .

Another example .

For example, the class is divided into groups . At least one person in each group , Otherwise, it will not form a group （ It doesn't exist ）. that , Each student belongs to only one group , This starts with mapping . And because each group must contain at least one student , There may also be multiple students , therefore ,“ Student ” aggregate （X） And “ team ” aggregate （Y） The relationship between , It's a kind of Full shot Relationship .

Students and groups （ Full shot ）： ① Or the group does not exist ; ② As long as this group exists , It must be pointed to by one or more students .

2.3 Double shot

It's a single shot , It's a volley again Mapping , It's called Double shot （ Britain ：bijection, Japan ： Full shot （ぜんたんしゃ））. “ all Y All the elements in are X The element in points to , And only by X An element of the . There is nothing that is not pointed to , And not by two or more X The element of the .”

for instance .

Most typical ,“ Student ”（X） With its “ Student ID number ”（Y）. Every student must have and only have one student ID number , So this is the mapping first . Again because , Every student ID number must be pointed to by only one student , There is no student ID number that is not pointed to by any student （ It is a full shot ）, There is no student ID number pointed to by multiple students at the same time （ It is also a single shot ）, therefore , This mapping relationship is a kind of Double shot Relationship .

Student and student ID number （ Double shot ）： One-to-one correspondence , Each other has only one element corresponding to itself in each other's set .

2.4 It is neither simple nor surjective , But mapping

There is also a mapping relationship , It is neither a single shot nor a full shot .Y There are elements in that have not been X The element of the , And pointed to Y The elements of , Not all of them are just one X The element points to （ There are many X The element points to ）.

for instance .

For example, go game . Yes “ chess player ”（X） and “ The board ”（Y） Two sets . A chess player plays chess on and only on one board , There are two players on a chessboard . however , Not all chessboards have players . such as , Yes 3 A chessboard , only 4 A chess player , that , There must be an empty chessboard , Not pointed by any player . Each player points to and only points to one board , This is a mapping . however , Not every chessboard has to be pointed by a chess player , Not every chess board is pointed to by only one player without exception , So this mapping relationship , Both It's not a volley , also It's not a single shot .

3、 Have you got it ？

Did you stop learning ？ Come on , Come on , Come on , Take out your little notebook and pen , Do two questions to practice ！

There are six questions . Write whether there is a mapping relationship between two sets , If there is , Write what kind of mapping relationship it is . requirement ： from “ Single shot rather than full shot ”、“ Full shot rather than single shot ”、“ Double shot ”、“ Not a single shot nor a full shot ”、“ Not a mapping ” Choose the right words to answer .

The answers will be posted in the comments section .

4、 Experience

I heard the question I often heard at school ：“ I know what this is for ？ Do I need it to buy vegetables ？” Um. , Of course you don't use it when you buy vegetables , You won't tell the stall owner , What is the mapping relationship between your dish and your people . however , for instance , When you design the database , You can discuss with your partners in the same group ,“ This field should be the primary key . Because the set of its values is bijective with the set of records in this table .” Not like this ： “emmm…… ah ？？ Primary key , I think it may be this attribute . Why? ？ because …… I can't say , Probably , Because of that , It's the only one . Ah No , It can uniquely identify a person ……emmm, A person has only one such attribute , This attribute only corresponds to one person ……emmmm, Maybe that's it , I'm not sure , I'm not sure , Think it over for yourself ！”

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