Introduction to Higher Mathematics - Lecture 9: Functions

A very useful type of relation between two sets is called a function, which we explore in this lecture. This is a somewhat different and more symmetric approach to functions from the one usually taken, instead defining it in terms of uniqueness and totality relations.

[sailtemporary]

@Bill Shillito I think you made a mistake in your slide at 3:07: "Every element on the RIGHT is mapped to by at most one elemement on the RIGHT." You used RIGHT twice.

[anishdhakal]

Error Notice: At 4:00, the last word of the first bullet sentence should be left not right. Thanks for the video.

[Myrslokstok]

These lectures are so good that I consider them mandatory for mathematical studying in higher levels of math. Thanks you are truly a great teacher!

[Stefabro]

thank you so much for posting these, you teach so well! so concise yet nothing important is left out. I tried so hard to learn thru just reading the textbook and it seemed impossible. but one video of yours made it seem so easy!

[tashp]

Thank you so much for these videos! I can't tell you how much they've helped in studying for my maths exams!

[Maistora]

This series is absolutely beutiful and amazingly done. Thank you so much for it! You have a great ability to teach and I hope you go more in depth in some topics or just make more math videos soon.

[usama]

i want to thank you so much as i have an exam on tuesday on the topic of sets, relations and grpups and your videos have made my life so much easier THANK YOU SO MUCH!

[clm]

after having done only analysis throughout high school algebra and calculus, coming upon this is fascinating. I had no idea how little of math I knew before coming upon this playlist!

[educmath]

Thanks for making this series of videos. As a university student studying maths, I really gain a lot from your videos. Not only learn something new, but also clear some of my misconception.

However, 27:30 "square roots of 9 is 3 and -3"
I think square root of x^2 is always equal to positive x (but not both).
When x^2=9, then x=sqrt 9 or -sqrt 9. That's why we can obtain 3 and -3 as answers. This is a common mistake made of many students. Therefore, I think you should emphasize on this.

[tomlee]

excellent series! Thank you. What you teach nearly matches the current course I am taking, which is called the foundation of modern mathematics.

[IslandForestPlains]

I rather like the terms "injective" and "functional" instead of "left-unique" and "right-unique" resp. The latter are new to me, and I find them confusing and counter-intuitive; as when I think about what which is which I end up with the opposite meanings! Left-unique sounds like a pair in the relation is uniquely determined by its left member so that just one connection-arrow leaves each element on the left; but this relationship is called right-unique.
Moreover, it requires one to think about domain and codomain in a left-right fashion which is why I would prefer "surjective" to "right-total" as well.

[BillShillito]

In general, for a nonzero complex number w, there are n solutions to the equation z^n = w, so we say that each z is an "nth root" of w. Look up "root of unity" on Wikipedia for more information. Something else to think about, btw, is that if we consider both square roots of a complex number such as -2 - 2i√3, we get both 1 - i√3 and -1 + i√3, neither of which we can really define consistently as "positive", which in complex analysis means we need a different way to choose the principal value.

[math]

please keep uploading these awesome vids.

[BillShillito]

This is actually something I'll be doing a video on soon. It is actually more correct to say that 9 "has two square roots", but that when we use the √ SYMBOL, we want the positive one. What I teach my students is that because of this, √(x²) is NOT just x, but rather |x| (absolute value of x). Then when they "√" both sides of the equation x² = 9, they get |x| = 3, which is WHY both 3 and -3 work as answers. This solves the question of exactly when we need the ± symbol and when we don't.

[warnford]

very helpful - many thanks - I quite like the axiomatic basis

[bryanchambers]

This lecture was easier than the last one for me. Probably because I am a physicist. Looking forward to number theory, I have always been curious to learn it.

[BillShillito]

Almost! We wouldn't say that S is total, we'd say that R is total over S. But yes, that would imply that either xRy or yRx.

[andypetsch]

4:53: The concepts of left-total and right total are different than the concept of "total" for a relation over a single set.
Because totality of a set S means, that for all x, y out of S it's true that either xRy or yRx?

[BillShillito]

Been having trouble having the time, but soon enough it will happen!

[NoNTrvaL]

24:22 it's actually f(f^-1(c)) = c.