description/proof of that identity map is bijection
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of bijection.
Target Context
- The reader will have a description and a proof of the proposition that any identity map is a bijection.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(S\): \(\in \{\text{ the sets }\}\)
\(id_S\): \(: S \to S, s \mapsto s\)
//
Statements:
\(id_S \in \{\text{ the bijections }\}\)
//
2: Proof
Whole Strategy: Step 1: see that \(id_S\) is an injection and a surjection.
Step 1:
\(id_S\) is an injection, because for each \(s_1, s_2 \in S\) such that \(s_1 \neq s_2\), \(id_S (s_1) = s_1 \neq s_2 = id_S (s_2)\).
\(id_S\) is a surjection, because for each \(s \in S\), \(id_S (s) = s\).
So, \(id_S\) is a bijection.
