Bijection Proof (a taste of math proof)

And in this video, we will show that f is a bijection by using the definitions.
Note: "one-to-one" is called "injective" and "onto" is also called "surjective"

#bijectionproof #mathforfun

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100/(1-x)

[Debg]

The girlfriend of one of my mates at uni was studying mathematics and she did a very interesting work about the maths behind origami and that stuff, supervised by the chief professor of the algebra department, known for being as brilliant as eccentric.

When she gave him a draft, he said 'it's perfect'. Then she asked if it didn't need a conclusions section and then he replied: 'we're mathematicians. Do you see that box at the end of the proof? That's the conclusion'.

[kais.]

your enthusiasm for the subject makes mathematics more enjoyable for me. Thank you!

[kwirny]

That's really cool, a box is even more satisfying than a q.e.d in my opinion

[ahzong]

That box at the conclusion means that everything you write for the proof is outside the box.

[nikolastevic]

Thanks, this really helped me understand functions better as I'm just learning them in school. Looking forward to video about inverse functions :)

[leeming]

You remind me of a really dear student I had first year college calculus,
He was so nice and loved mathematics,
He had limited patience but he helped me a lot,
even though I was slow.
He would solve problems very interesting out of box ways,
He sometimes had to explain to teacher! haha

[andrewm]

I watched this before am exam and met a question in that exam where I applied the knowledge. I could recall him explaining. It felt amazing. Keep up

[paulhaso]

Your silly jokes and gestures just make my day

[BrainGainzOfficial]

Loving this kind of content lately! Keep it up! 🧠💪🏼

[justfahid]

y'all be making it more hard than it needs to be

[anabang]

First week physics/math students be like:
Wow that's difficult, better not learn it and make fun of people using those terms.
*exmatriculates furiously*

[Red-otlv]

Dont typically comment on videos but man i love your enthusiasm!

[Albkiller]

Discrete math flashback coming out... lol

[jayapandey]

The coolness in Real Analysis diverges.

[japotillor]

I enjoyed analysis, being a Latin geek, I use QED, quod erat demonstratum (though the box is cool too). I taught my HS students this...they hated me :p (our analysis teacher made us promise if we taught algebra that we would cover injection and surjection)

[emperorpingusmathchannel]

Well it is pretty self evident why *this* function is a bijection.


Define odd n= 2k-1 where k belongs in the set of the naturals
Define even n=2k
Note: all integers below 0 equal to -n (ie: -abs(z)=-n)
Note: all integers above 0 equal to n (ie: abs(z)=n)


g(k)=2k-1 and h(k)=2k
There exists no integer in the set g(k) such that g(k)=h(k)
since 2k-1=2k is untrue because it requires -1=0 to hold true.


If you plug 2k-1and 2k in to f(n) you'll get
f(2k-1)= -n

f(2k)=n


Since the domains are mutally exclusive and the ranges are mutaully exclusive, you have sufficiently proven that it is a bijection.

[angelmendez-rivera]

f(n) can be expressed as [(-1)^n][n + (1 - (-1)^n)/2]/2 = [(-1)^n][(2n + 1)/4] - 1/4. The inverse function, call it g(n), can be expressed as sgn(n)[2n + (1 - sgn(n))/2] = 2|n| + [sgn(n) - sgn(n)^2]/2.

Can you find an explicit formula for a bijective function between the natural numbers and the rational numbers?

[thedeg]

Killer video! Well explained and Helped me a lot :)

[angrycat]

This was very helpful to me as I’m starting to self study analysis. Thanks!

[handymandycraftsfashion]

I learned more from a 11 minute video rather than one and half an hour of my class😇