 In this final video on Boolean algebras, I wanted to describe very quickly an application. That is an application beyond mathematics. It turns out that the Boolean algebras are very important in the construction of electrical circuits. In fact, the modern digital computer is based upon algebra of Boolean algebras. Why is that? Well, electrical circuits can be designed to be able to create the simplest of all Boolean algebras. That is Boolean logic, where we just take two values, zero and one. From the logical side of things, you can think of zero as false, one is true, and therefore join is and in that situation, excuse me, it's or in that situation, and is meet over here, for which if you take false or false, that's false. If you take false or true, that's true, true or false is true, and true or true is true. Similarly with and, if you take false and false, that's false, false and true is false, true and false is false, and true and true is equal to true. That's just our most fundamental Boolean logic algebra. But you can also work with electrical systems here, where you think of not as statements, but you think of switches or gates of some kind. Think of a light switch. A light switch has two possible options. A light switch is either on or off. If it's off, we could say that zero. If it's on, we'd say one, or you could think of it as like a gate. A gate is either open or closed. So if we ask the statement, is the gate open? We can think of the true statement, one if it's open, zero if it's closed. And so this idea of a gate or a switch being open or closed has applications towards circuits. So imagine we have some type of circuit over here, and then there's this copper wire that comes out of it. And then you have some other part of the circuit over here. And imagine there's some type of gate in the middle here. So if the flow of electricity is going in this direction, if this gate is open, then the electricity will flow to the other side of the circuit. But if the gate is closed, then you can't get through. And so the electricity can't flow through. So open gates flow, closed gates don't flow. And so we can actually construct these two operations using just the flow of electricity. So imagine you have some circuit over here, and then without describing it too much more, I'm just gonna say you have some circuit capital A. And then you have some gate that comes along the way. We'll call it little A. And then you have this other part of the circuit B so that you have flow. And honestly, it doesn't really matter if the flow of the circuit goes from capital A to capital B or vice versa. The thing is if A is open, right? If it has a value of one, then we can flow through the circuit. We can flow through that line to the other part of the circuit. And if it's closed, then like I said, there's no flow. That's gonna happen between circuit capital A and circuit capital B. So the gate can capture that thing. So then what we can do is sort of the following idea with regard to meat. This one's actually much easier to describe. If you have some circuit A that's flowing to B, okay? And let's put in two gates in this situation like so. So we have gate A and we have gate B for which these could be open or shut, all right? And so let's say that the electricity is flowing from capital A to capital B. Well, if lowercase A is shut, then you can't flow through this thing. You're gonna get blocked. In which case that's kind of something like this, right? Regardless of whether B is open or not, if A is closed, you can't flow through, right? So if A is closed and B is closed, then there's no flow. If A is closed but B is open, there's still no flow because you can't get through A, all right? So that's kind of capturing this idea right here. If A is closed, then the conjunction of the two gates is as if it was closed. But then conversely, if B is closed and A is closed, then there's no flow. But if A is open, so you can flow through little A, but then you get blocked at B, so there's no flow there as well. And so again, the composite of these two gates is that if B is closed, it doesn't matter what A is doing, there's no flow. And then finally, if both gates A and B are open, then the electricity can flow through this wire and we get that you get flow through the combined gates only when they're both open. So if you put these, if you sequence these circuits together, a so-called series, if you sequence these together to gates, their series is in fact a and operation, like so. So by sequencing them, we can construct this and operation. Can we do something similar for or? The answer is yes, of course. So again, we take these two blob circuits. We don't care what they are, circuits capital N, circuit capital B. And again, assume A is flowing towards B, but it doesn't make much of a difference there. And so then you come along and let's suppose we take on some type of parallel circuit. So these two circuits that are running, well, these two lines in the circuit that are running parallel to each other. Or again, we can assume the flow is going from capital A to capital B. In this situation, if one is open and B is open, then we can flow from both sides, there is flow from A to B. That matches up with this statement right here. If A is open, but B is closed, so there's a blockage down this path, but we can still flow through the A path and that would be something like this. Okay, if A is open, but B is not, you can still flow from capital A to capital B. And then conversely, if A is closed, but B is open, then you get blockage on little A, but you can still flow through little B. And so that would be this situation here. Little A is closed, but little B is open. You can flow the electricity from capital A to capital B still. The only way that you block the whole thing is if you get zero, zero for both gates, A and B. So if these are both blocked, then there's no way from for A capital A to flow to capital B. And so we get something like this. So using this idea of parallel circuits for, or and using series for and, we could actually create these operations just using flow, right? This is just flow for which you could do this with like water pipes if you wanted to. So we could build a water-based computer. I don't think anyone can go around doing that. Electrical computers seem to be pretty effective, but that is we could build these type of gates, these type of logic gates, as they're sometimes referred to, to construct the logical operations of and and or, okay? For which you can also do things like compliments for which the compliment is always closed when the circuit, when the one gate is open and vice versa. And you can start building more complicated gates by joining these together. So using parallel and serial circuits, you could combine them together to make any possible Boolean algebra statement. So it doesn't matter how convoluted you want it to be. If you have something like A or B intersect C compliment, join, I guess that's a meat, D join E or something, whatever, you know, any Boolean statement like this can be built into some type of combination of series and parallel circuits. And so using flow of electricity and honestly the flow of anything, we could compute any Boolean expression based upon the inputs. And this can be done automatically because after all, no one has to think about it. Once the equation, once the function, you know, once the logic gates are built, we just have to flow electricity through based upon our inputs. We flip switches when we have different inputs and then automatically changes the output. The computation is automated. No one has to think about it. No one has to compute it because we built an actual object that'll compute the outcome of that Boolean expression as long as we're able to change the switches, right? If we can flip some switches and boom, it calculates automatically. And so that's a pretty cool idea. This is the birth of modern computation, at least in the digital sense, all right? And so the digital computing is based upon this idea. So let me give you an idea how one can build a calculator using such a thing here. Every number, every integer, we'll start with that. Every integer, we won't worry about negative numbers and we won't worry about decimals or anything like that. So we won't get into floating points at the moment, but if we just want to look at an int, just an integer right here, every integer can be represented in binary notation, right? So given any number in, we can write this as a sum where we have some coefficient C i times two to the i, where i ranges from actually zero to infinity, right? All but finally, many of these C i's will be zero. This series will always converge, but some of those C i's could be one and every number can be expressed uniquely in that fashion. Like if we take, for example, the number 12. The number 12 is eight plus four for which eight is two cubed and four is two squared. And so if we think of it in binary notation, we have one times two to third plus one times two squared. You're gonna get zero times two to the first and zero times two to the zero. Putting all that together in binary notation, 12 is the same thing as 1,100. And so we'll put the two down there to represent base two. And we can do this with something else, right? You know, like the number 13 would equal 1,101. You know, 14 would equal 1,110. We could keep on going, right? Every integer can be uniquely decomposed into binary representation, okay? And so because of this binary, it really comes down to, can we add together in binary one digit at a time? That's all you have to worry about. And so if you're adding together these two bits, so you have some bits, B1 plus B2. I want you to be aware that when you add them together, and well, the table, the Cayley table looks something like the following. We're adding together here, so you have two possible options for each bit, zero and one. Well, if you take zero plus zero, that's gonna give you zero. Zero plus one is one. And one plus one is equal to 10 in that situation. For which if you look at the join operation, right? This exactly agrees with what you get there. So zero plus zero is just zero, join zero. Zero plus one is just zero, join one. Plus zero is the same thing as one, join zero. That's exact same. There is disagreement though on this right here. With one, join one, you want there to be a one. So how do we detect when we should carry, right? Because after all, when you add the unit digit, right? It should be a zero. How do you carry it though? Like we should carry the one when we do the arithmetic there. Notice when you look at the and operation and only produces a one when they're both ones. So basically you can build binary arithmetic by combining ORs and ANDs because you first check, are they both ones? If you use an AND circuit, an AND gate, then that'll detect whether they're both ones or not. In which case, if that is, you know the answer is gonna be 10, okay? If it's not, then you run those, you run them through an OR circuit and it gets you the exact value. So we can add two binary bits together. And then by recursing this process, we can do more complicated, complicated arithmetic and voila. We've now reached the birth of digital computing. Clearly things can get much more complicated than this, but I just wanna give you just a very basic idea of how bull and algerists have huge ramifications to electrical circuits and of course digital computing. Now, one last comment I wanna say before ending this video here is that when it comes to a digital computer, it's based upon the fundamental idea that you have two options and you have either OR or OFF. A switch is either ON or OFF. Those are the only two options, zero or one. A gate cannot simultaneously be open and OFF. And that's the basis behind these so-called bits. You only get zeros and ones. Now, in the realm of quantum computing, we now get the notion of a Q-bit which we allow for other possibilities. Like it's possible that our gates in a quantum computer can be simultaneously ON and OFF at the same time. That might seem kind of weird when you think of it from a digital computing point of view, but quantum computers are pretty interesting in that regard because they think that it doesn't follow this same Boolean algebra. Now, are they Boolean algerists? That's a great question. Alas, we are at the end of our series and so we're not gonna go into quantum computing. Just wanted to illustrate how Boolean algerists do have ramifications for digital computing, which many of us are familiar with. And so this does bring us not just to the end of lecture 39, this brings us to the end of our lecture series math 4230 abstract algebra two. If you've been along for the whole time, I very much appreciate it. If you're just here just for this one video, that's cool too, that really is. Check out some of the other videos if that's you. So as I end this lecture series, I just wanna thank everyone for coming with me on this journey. I hope you learned a lot about abstract algebra, not just the theoretical aspects, but many of the applications as well, much like we saw in this video right here. Many people think that pure math and applied math have no overlap whatsoever, but that's not true with regard to their Venn diagram. There are some impressive things that exist between pure mathematics and applied mathematics. Abstract algebra is there too. While abstract algebra typically is viewed as a pure mathematical subject, there are many applications of pure, of abstract algebra inside of applied mathematics, such as electrical circuits that we saw in this video. So I hope that you appreciate the combinations we saw between the theoretical and the applied aspects of abstract algebra. And whether you are a pure mathematician, an applied mathematician, or just a lover of mathematics, I hope you can see the beauty and the utility of abstract algebra. And I hope this lecture series will encourage you to study more abstract algebra in the future. As always, if you do have any questions on any of these videos you have seen, please post them in the comments below. We'll be glad to answer them. If you've learned anything about abstract algebra, please like these videos, subscribe to the channel to see more videos like this in the future. And if there's specific topics that you would love to see that currently aren't covered in this library of videos, please recommend them in the comments. And I will be glad to do it as soon as I have the opportunity and the interest to do so. All right, bye everyone.