 As we end lecture 11 with this video, it's appropriate for us to reflect on what we've done so far in these first 11 lectures. So you'll recall that our first few lectures introduced us to the notions of set theory, just very elementary stuff, just the basics of set theory. And that was our first unit. Now we're in the middle of a new unit about combative torques. And these notions of sets and counting sets are helping us understand better advanced mathematics. Now, in order to understand advanced mathematics, we have to also understand the notions of logic. And so every lecture in this series so far has had a lecture that teaches something about logic. And many of those lectures have been focusing on what we refer to as Boolean algebra or Boolean arithmetic, sometimes called Boolean logic. That is this algebraic manipulation of statements. And so we have these operations of conjunctions, disjunctions, conditionals, negation. And so these are very much like arithmetic with logical statements. It's important to be aware of the reasons that we have studied logic so far. Honestly, there's probably three main reasons why in a class like this one, math 3120, we study logic. First, the truth tables we study tell us the exact meaning of the words such as and or not and things like that. The second reason I would say is that the rules of inference, which we'll talk about those in the next lecture, the lecture 12, these rules of inference provide a system which we can then produce new mathematical truths from already known, already proven mathematical truths. And then the third reason is that logical rules like De Morgan's laws, which we talked about in the previous video help us correctly change certain logical statements into other logical statements. Hopefully more useful statements which have the exact same meaning because they're logically equivalent. So this would be like in an algebra or arithmetic class doing something like, oh, I could take X plus one and I can foil it as X plus two times by X by two. And so you end up with something like OX squared plus three X plus two. This is algebraically equivalent as an expression. Maybe this is more useful than the factorization or sometimes we go the other way around. So what we've done with logical laws and truth tables is understanding how we can do algebra with logical statements. These are all very important things for us as we transition into advanced mathematics. Logic is the common language that all mathematicians use. So we must have a firm grip on logic in order to write and understand mathematics. After all, math 3120 is a communications class. We have to be able to read, write, speak, listen, mathematics and we cannot do that if we don't understand the, we don't understand logic. If we think of mathematical communication like other languages, then in order to read, write, speak, et cetera, we have to understand grammar. So grammar in language is important. And for us, this is kind of what logic is doing. Logic in some essence is the mathematical grammar inside of our language. And grammar, of course, serves as a fundamental role in any spoken or written language. In much of the same way, logic's place is fundamental in mathematics, but much like grammar, logic's place is actually in the background of what we do. It's not the forefront. Our study of logic going forward in this lecture series will be more focused on proof patterns as opposed to the arithmetic of logical statements that we've seen thus far. And this is analogous to how one might study a language, might study writing in a language like in English. Grammar is a part of it and we do have to understand the grammar, but when it comes to writing, it's so much more than grammar. And grammar is usually just a minor thing we worry about every once in a while. We think about when I write, do I write with clarity? How well composed is my writing? Is my writing correct? Is it clear? These are the types of things that we worry about. Grammar is only one small little part. It's important. If you don't follow the rules of grammar, then your language, your writing will be garbage, but be aware that grammar is just one of the small things we consider. And so therefore in the next lecture, lecture 12 and in future lectures here, when we dedicate time to talk about logic, instead of the Boolean logic we've talked about so far, we'll pivot towards more logical inference, which we'll talk about in the next lecture, and then the consequences of logical inference, which is how do you argue mathematically? How do you prove something mathematically? And so the structure of mathematical proof is what we will see going forward. We'll continue to study mathematical topics like combinatorics and sets and future videos as well. And so like I said, this does bring us to the end of lecture 11. I appreciate you watching this video and all the other videos in this lecture series. If you learned anything from these videos, please like them, subscribe to the channel to see more videos like this in the future. And if you have any questions whatsoever, please post them in the comments below and I'll be glad to answer them as soon as I can.