Скачать или смотреть GF3 Inverse of Functions & Bijective maps

What is the inverse of a function? Theorem & Proof: A function has an inverse if and only if it is a bijection. Connections to Combinatorics & Group Theory. Subscribe ‪@Shahriari‬‪@Shahriari‬ for more undergraduate math videos.
00:00 Introduction
00:20 What is a function? What are 1-1, onto maps? (   • GF1 What is a function? What do 1-1 and on...  )
01:40 What is function Composition? (   • GF2 Function Composition & 1-1, onto maps  )
02:18 Theorem: The composition of two bijections is a bijection
02:47 What is the Inverse of a Function?
03:13 Definition: Identity Map
03:44 Inverses Visually
04:21 Definition (formal): Inverse of a Function
05:09 Examples: Inverse of functions f : R to R
08:45 Theorem: A function has an inverse if and only if it is 1-1 and onto
09:14 Proof of the Theorem
13:16 Corollary: The inverse of a function (if it exists) is itself 1-1 and onto
13:53 Example: bijections in Combinatorics show that two sets have the same size (   • CO12 Balls & Boxes Counting Problems  )
18:12 The set of bijections from a set to itself forms a group (   • AG03 Permutations and Bijections form a Gr...  )
20:30 Elements of S_3, the symmetric group of degree 3 (   • AG04 What is the Symmetric Group S_n? Cycl...  )

The vocabulary and the results are for general functions from one set to another (as opposed to just real valued functions on the real line) as encountered in courses in Linear Algebra, Combinatorics, Analysis, or Abstract Algebra.

One of a series of lectures by Shahriar Shahriari on basic mathematical concepts used in undergraduate college mathematics.

The Series on Functions is as follows:
GF1: Functions, 1-1, onto Functions    • GF1 What is a function? What do 1-1 and on...  
GF2: Function Composition    • GF2 Function Composition & 1-1, onto maps  
GF3: Inverse Functions (this video)    • GF3 Inverse of Functions & Bijective maps  
GF4: Cardinality (uncountable sets, Cantor's diagonal argument, and the Continuum hypothesis)    • GF4 Cardinality--Countable & Uncountable Sets  

Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA
Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and six time winner of Pomona College's Wig teaching award.