When we grow up and take math courses, we tend to ask: "Why is that true?" Most of the time, the teacher tells you something like "because that's the way it is." In this course, we reintroduce all of mathematics in the perspective that we should have. We answer and PROVE the "why" in our lives. The greatest part is that we are dealing with truth - something that science or any other subject can talk about. Everything proven in this course is absolute truth and cannot be denied.
Many students get their first exposure to mathematical proofs in a high school course on geometry. Unfortunately, students in high school geometry are usually taught to think of a proof as a numbered list of statements and reasons, a view of a proof that is too restrictive to be very useful.
Learn How To Prove It Today!
In this course, we learn how to prove amazing mathematical facts:
The proofs in this course may sound extremely simple, but if I were to ask you to prove 1+1=2, could you do it? It takes 362 pages to prove this fact! Well, there are better methods today, but the process of proving things is more fun than you can imagine!
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Welcome to my course! I am excited to show you a new perspective on deduction and logic.
See how to compare two logical systems and determine equivalences.
See an example of how we can prove basic equivalences in logic theory.
Though we know how to prove these statements, learn how we can simplify more intricate logical sytems using a few easy rules.
Learn about the first and only object needed in all of logic and mathematics.
Watch and see how these sets can act a lot like logical systems.
Realize the misunderstandings of the implication and how unicorns are in fact purple!
Sometimes we want to say (p->q)^(q->p), but the biconditional takes care of this common logical relation.
In this lecture, we prove basic facts about set theory without using truth tables.
Understand existence and universal quantification.
Understand everything about the empty set!
How can you take an English statement and determine the opposite of that statement?
Test your knowledge of quantifiers and the ability to negate.
Thought we were done with sets? In this lecture, we learn more about operations on sets.
Sometimes, it's not possible to prove a mathematical fact, so we need more proof techniques.
This is one of the most common proof techniques. Prove an implication by assuming the implication fails and that this implies a crazy contradiction such as 1+1=fish!
Mathematical induction is a very different and uncommon approach but sometimes is very necessary!
Forget everything you learned about the x-y plane in Algebra class. In this course, we reinvent this idea in a more concrete way.
What does it mean for two things to be equal? Sometimes this definition can change, but there are some rules about when two things are equal that have to be satisfied.
Forget everything you learned about functions. We approach this idea very much algebraically, however we give a new perspective.
There are some properties of functions that may seem useless but end up being used all over mathematics in very unique ways.
There is yet another way to compare things. Sometimes we want to say certain things are "bigger" or "better" than other objects. In this lecture, we state the rules needed to make this possible.
There are two infinities!
Let's start proving some weird things about infinity...
It turns out that this giant infinite set is tiny compared to other sets.
Learn about one of the biggest sets in set theory. We prove that the set of real numbers is, in fact, uncountable.
Congrats on finishing this course!
Math can be hard, but it doesn't have to be!
I love what I teach. I feel like math has a negative connotation to it, and that it is the teacher's job to build enthusiasm and interest through their own passion for the subject. Right now, most students take their math classes just to get the degree requirements - and I respect that - but I also want the student to enjoy what they are learning. This can be hard to do, but I am willing to try my best. When students hit a wall in their mathematics career, then they need someone to help them back up. My goal is to be that person. I have seen how many professors teach, and there are many styles that I like to incorporate. I like to show math in a different and interesting perspective that hopefully is also applicable.