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).
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Here is another proof using diagram chasing that function composition is associative: • Function Composition i...
If you are looking for other videos to help with proof writing skills, check out this playlist: • Proof Writing
This animation arose out of a class I am teaching about abstract algebra using the textbook from Tom Judson:
abstract.ups.edu/
#manim #function #injective #surjective #composition #visualmath #math #mtbos #visualproof #settheory #introductiontoproofs #associative #functioncomposition #universalstatement #functiondiagram
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[@Ninja20704]

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.

[@MathVisualProofs]

That’s good! It’s really the best way to make sense of those types of proofs.

[@LittlePaimon]

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

[@MathVisualProofs]

👍

[@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!

[@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

[@MathVisualProofs]

Haven’t thought of doing them yet. I did use the software to show certain concepts in real analysis a few years ago but I was brand new to the software so I wasn’t doing things quite right.

[@alexandrashvydun8726]

appreciate the animation!

[@MathVisualProofs]

👍

[@ileoliang]

very nice

[@MathVisualProofs]

Thanks

[@DidarOrazaly]

How call this theorem?

[@bluepavilion82]

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

[@MathVisualProofs]

I don’t. I bet you can work it out using this as a model though …

[@hashimabbas3977]

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.

[@MathVisualProofs]

I'll see if I can do a follow up, but I bet you can do it on your own! Also, this video doesn't seem to be as interesting to people as others by clickthrough rate.