Function Proof via Diagram Chasing (g composed f injective and f surjective implies g injective).

This is a proof, with a visualization via diagram chasing, of a standard exercise about showing that a particular function is surjective (onto). In particular, if f is surjective and the composition g composed with f is injective, then we can show that g is injective. The video includes three "visual definitions" for the terms function, injective, and surjective, and the definitions are utilized when they are needed in the proof. The final proof is written in a way that I think would be acceptable for submission in a class (using words and not symbols and fully justified). Implicit in this explanation is the style of proof required when proving a universal statement (here we use a definition of injective that takes the form of a universal statement followed by a conditional).

This animation arose out of a class I am teaching about abstract algebra using the textbook from Tom Judson:

#manim #function #injective #surjective #composition #visualmath #math #mtbos #visualproof #settheory #introductiontoproofs #associative #functioncomposition #universalstatement #functiondiagram

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[Ninja]

Very good visualisation. My classes always try to explain function properties using the diagram chasing, as well as stuff like the conditions for an inverse and composition to exist.

[LittlePaimon]

The video presentation is very intuitive, and it seems the order of differentiation for composite functions follows a similar pattern.

[Bedoroski]

Great visualization.
Btw do you intend to make visual understanding of some theorems in Real Analysis? As an undergraduate that lacks proof background this class hurts me

[tcaDNAp]

Did you hear about the mathematician that couldn't prove f was surjective? **Epic** fail...

Thanks for helping me understand a little more about diagram chasing than jokes from category theory!

[hashimabbas]

Why dont you make videos of all the theorem.
It will help us to understand maths very effectively, including the 2nd one in the video.

[bluepavilion]

do you have proof solution for the theorem at the end of the video?