WEBVTT 00:00.440 --> 00:01.320 Hello everyone. 00:01.320 --> 00:03.640 Welcome to this lecture on Boolean algebra. 00:03.880 --> 00:09.120 This is a fundamental concept that powers modern computers, digital electronics and all logic based 00:09.120 --> 00:09.720 systems. 00:10.120 --> 00:16.480 This topic lies at the heart of computer organization, digital circuit design and decision making in 00:16.480 --> 00:17.480 software logic. 00:17.880 --> 00:21.520 Now let's begin with the deeper understanding of its origin and significance. 00:22.120 --> 00:32.640 In the 19th century, an English mathematician named George Boole developed a new branch of mathematics 00:33.360 --> 00:37.840 to model logical thinking using symbols and equations. 00:38.480 --> 00:45.440 His central question was can human reasoning be expressed mathematically? 00:45.680 --> 00:55.240 His answer formed the foundation of Boolean algebra, a system where variables only have two possible 00:55.240 --> 00:56.040 values. 00:56.760 --> 01:08.410 It's either true or false Represented true by one and false by zero in computing this binary system 01:08.450 --> 01:09.050 load. 01:09.090 --> 01:12.930 Logic problems to be expressed in a formal mathematical way. 01:13.530 --> 01:14.890 The impact of this was. 01:15.570 --> 01:20.930 Far reaching, laying the groundwork for logic control control systems. 01:20.970 --> 01:25.450 Digital logic and eventually the structure of computer processors. 01:26.090 --> 01:28.730 Now let's draw a clearer comparison here. 01:29.410 --> 01:42.290 In traditional algebra, traditional algebra operates on real numbers like nine, 15, or 996, but 01:42.290 --> 01:45.570 the Boolean algebra operates on a binary values. 01:46.290 --> 01:49.010 It is either 0 or 1, right? 01:49.450 --> 01:50.730 A or b. 01:51.570 --> 02:01.630 The traditional gabra uses arithmetic operations like plus, minus and for example, modulus and so 02:01.630 --> 02:02.350 on and so forth. 02:02.990 --> 02:10.550 But the Boolean algebra uses logical operations, for example, and or not. 02:11.310 --> 02:13.470 And this goes on the third. 02:13.510 --> 02:25.620 In traditional gabra we have infinite values allowed like you can combine, for example 999532 4 or 02:25.620 --> 02:27.230 6 seven. 02:27.230 --> 02:30.910 And this this is goes like this is infinite basically. 02:31.670 --> 02:38.110 And in Boolean algebra you can use only two possible values. 02:38.310 --> 02:41.550 It is either true or false. 02:42.310 --> 02:46.830 In computer we call true one and false zero. 02:47.230 --> 02:54.270 But for example in your own logic you can call the true t or f a or b. 02:54.950 --> 02:56.390 Yeah, you get the logic here. 02:56.390 --> 03:03.920 You can only have two values AWS, which can represent either true or false and result. 03:04.000 --> 03:07.320 Result is what in the traditional algebra. 03:08.080 --> 03:14.000 The result is a numerical computation and the Boolean algebra. 03:14.440 --> 03:18.120 The result is logical decision. 03:19.640 --> 03:33.600 As a beginner in Boolean logic, firstly we will learn about and or and not and returns one only when 03:33.880 --> 03:34.480 both. 03:37.840 --> 03:40.240 Both inputs are one. 03:43.480 --> 03:56.640 So one and one equals 1 or 1 and zero equals zero. 03:57.360 --> 03:59.690 It will only return Turn one. 04:00.530 --> 04:11.130 If at least one or both inputs are one, but in or returns one if at least one input is one. 04:11.450 --> 04:14.610 So if we have zero. 04:16.770 --> 04:20.290 And one, we have one. 04:20.690 --> 04:27.570 If we have zero and zero, we have zero. 04:28.330 --> 04:34.770 And the node inverts the input, turning one into zero and zero into one. 04:35.170 --> 04:45.450 So not one equals zero or not zero equals one. 04:45.490 --> 04:46.010 One. 04:47.010 --> 04:53.570 The Boolean algebra provides a compact and efficient way to represent truth conditions that occur frequently 04:53.570 --> 04:56.490 in programming, hardware control, and logical automation. 04:56.490 --> 04:59.010 So this is this was just the beginning to. 05:00.590 --> 05:02.390 meet you with what you will learn. 05:02.390 --> 05:11.470 In this section of our course, we will explain these conditions and values more deeply and detailed 05:11.470 --> 05:12.670 in next lectures. 05:13.430 --> 05:15.110 So don't worry. 05:15.790 --> 05:21.830 Hello everyone, I'm Typhoon here again and in this lecture we are going to explore the basic boolean 05:21.830 --> 05:26.470 operators the and or and not. 05:27.070 --> 05:33.350 These are fundamental tools in digital logic and they are used in nearly every electronic device you 05:33.350 --> 05:38.630 can think of, from computers and smartphones to microwaves and traffic lights. 05:38.870 --> 05:43.750 If you're just getting started with a computer organization or digital systems, understanding these 05:43.750 --> 05:47.830 operators is your first big step and I will guide you through each concept. 05:47.830 --> 05:53.710 Explain how it works, show you the symbols, provide truth tables, and walk you through how this is 05:53.750 --> 05:56.230 implemented in an actual electronic circuit. 05:56.950 --> 06:03.720 So what is a Boolean operator A Boolean operator is a special kind of function that works on 1 or 2 06:03.760 --> 06:05.120 binary values. 06:05.120 --> 06:16.440 Binary values are simply 0 or 1, where zero usually means false or off, and one usually means true 06:16.480 --> 06:17.160 or on. 06:17.440 --> 06:25.150 Now the values that Boolean operators act on are called operands, and the result is also either 0 or 06:25.150 --> 06:25.600 1. 06:26.320 --> 06:33.040 Now Boolean operators are essentially act on are called operands. 06:33.800 --> 06:38.080 So to make things clearer we will use truth tables. 06:38.120 --> 06:45.760 A truth table is a tool used to show how the output of a logic operation changes based on all possible 06:45.760 --> 06:47.120 combination of inputs. 06:47.120 --> 06:52.600 But before we jump into the operators themselves, let's warm up with a real world example from binary 06:52.800 --> 06:53.600 arithmetic. 06:54.280 --> 07:07.450 So the third example we will do here is we will create x y and we will also have carry and we will have 07:07.490 --> 07:08.330 some here. 07:09.050 --> 07:13.410 So let's draw the lines here one. 07:14.250 --> 07:22.650 And now we have the two binary values x and y and add them. 07:22.650 --> 07:26.170 When you add binary numbers there are two things to think about. 07:26.290 --> 07:28.370 The sum we have here. 07:28.690 --> 07:32.770 This is the result of the addition and we have the carry here. 07:33.010 --> 07:37.850 If both bits are one, we carry a value to the next position. 07:37.850 --> 07:41.610 Now here's how the result looks in a truth table. 07:42.010 --> 07:46.250 So here we will have zero and zero. 07:46.610 --> 07:48.410 We have zero carry as well. 07:48.410 --> 07:55.490 And the sum is what the zero we have 0101. 07:56.090 --> 08:02.430 We have zero carry but the sum is one because of why is one here? 08:03.190 --> 08:06.630 Or else we have one and zero again, same. 08:07.390 --> 08:15.990 But if you have two ones, we also we will have another carry and the sum is going to zero. 08:16.030 --> 08:20.150 But we will have just a small one here which we will not talk about that right now. 08:20.710 --> 08:24.430 Now this is how binary addition works inside a digital circuit. 08:24.670 --> 08:31.830 And now every row shows one possible input combination and what the circuit should output in that case. 08:31.830 --> 08:36.630 And this as you can see, if we have both ones we will have the sum of zero. 08:37.350 --> 08:42.390 Now we will return the binary addition later when we design circuits called adders. 08:42.390 --> 08:45.910 But for now let's look at the core logic operations. 08:46.590 --> 08:51.630 Logic gates are physical electronic components that perform Boolean operations. 08:51.670 --> 08:56.190 Now think of them as a small building blocks inside a computer's processor. 08:56.230 --> 09:02.680 Now each type of logic gate has a unique shape and behavior, and it performs one specific logic function 09:02.680 --> 09:06.840 like and or not. 09:07.440 --> 09:12.240 And in the next lecture, we'll cover each operator in detail, including its definition, the truth 09:12.240 --> 09:13.760 table, and the circuit symbol. 09:14.480 --> 09:16.800 But now we will start with the end. 09:18.640 --> 09:22.520 So we have the and operator also called the contraction. 09:23.320 --> 09:25.080 The type is binary operator. 09:26.040 --> 09:28.040 All of these are binary operator. 09:28.720 --> 09:32.600 How we can tell that it's a binary operator because it uses two inputs. 09:32.600 --> 09:36.240 It is either 1 or 0. 09:36.560 --> 09:38.200 So it's a binary operator. 09:38.840 --> 09:42.360 And the symbols are symbols are this. 09:42.920 --> 09:47.040 Now this is a common in logic and also common in programming. 09:47.800 --> 09:49.520 Is this one just a dot. 09:50.240 --> 09:51.960 Or you can also just write. 09:52.000 --> 09:57.080 And this is a symbol symbols of ands here. 09:57.770 --> 09:58.250 right. 09:58.970 --> 10:04.770 The output there is rule is the output is one only if the. 10:05.010 --> 10:06.490 Both inputs are one. 10:06.730 --> 10:09.410 So let's do another example here. 10:10.170 --> 10:20.810 We have a b value and we have x and or we have a and b. 10:21.570 --> 10:26.170 So what we will have here we will just two three and four. 10:27.370 --> 10:31.970 So if we have a zero and b zero we will get zero. 10:32.210 --> 10:35.890 If we have a0 b1 we will get zero. 10:36.330 --> 10:40.330 If we have a1 b0 we will get zero again. 10:40.330 --> 10:45.930 But if we have A1B1 we will get one. 10:46.250 --> 10:49.810 Now this is how and operator works. 10:49.930 --> 10:52.730 So imagine two switches connected to the light. 10:52.770 --> 11:00.630 The light turns on only if both switches are on or for example, like this. 11:00.630 --> 11:01.950 This is a light bulb. 11:01.950 --> 11:07.670 Here we have two switches and this is battery right. 11:07.990 --> 11:11.070 This is just how and works. 11:11.830 --> 11:16.630 And if you are designing computer circuits and has a circuit here. 11:17.350 --> 11:23.190 So for example x and y the circuit here is what. 11:23.870 --> 11:28.150 Now this is a circuit of and operated. 11:28.190 --> 11:31.510 Now this is how you represent and in circuits here. 11:32.230 --> 11:35.470 Now the gate is shaped like D as you can see here. 11:35.510 --> 11:44.470 Now the behavior is similar to multiplication and regular math uh, which is uh one by one equals one. 11:45.230 --> 11:46.950 Remember this is just like that. 11:47.430 --> 11:50.750 And all other combinations are zero here as well. 11:50.790 --> 11:52.430 But just this one. 11:53.110 --> 11:59.840 And we also have the Or Operator, also called as disjunction. 12:00.640 --> 12:03.320 The type is binary operator as well. 12:03.720 --> 12:14.360 The symbols is V in logic, plus in math or just or in programming. 12:15.200 --> 12:19.720 The rule is output is one if at least one input is one. 12:20.440 --> 12:37.800 For example, we have a b and x or a or b, so if we have a zero and b zero, we will have what zero? 12:38.160 --> 12:43.600 If we have A0B1 we will have one. 12:44.240 --> 12:49.560 If we have a one and b zero we will have one. 12:49.760 --> 12:56.290 If we have A1B1 we will have one as well. 12:56.330 --> 13:01.450 Now you can see we got three and just one zero here. 13:01.650 --> 13:05.370 Now think of the two doors that led to the same room. 13:05.410 --> 13:07.530 If either door is open, you can enter. 13:07.570 --> 13:18.050 Now that's how or works and how you represent or in logic is x, y, for example. 13:19.810 --> 13:25.010 And this is how you represent or in, uh, circuits. 13:25.410 --> 13:27.570 Now the gate has a curved shape. 13:27.850 --> 13:29.490 It behaves like an addition. 13:30.050 --> 13:37.730 Uh, as we mentioned here, uh, zero plus one is one, and one plus one is still one. 13:38.290 --> 13:40.970 So we don't carry anything in the Boolean logic. 13:41.730 --> 13:45.170 Now we have the not operator. 13:45.210 --> 13:49.690 Now I would say that not operator is easiest among these. 13:50.010 --> 13:57.630 Now the not operator, also called negation, is represented in a unary operator. 13:58.790 --> 14:00.230 It uses only one input. 14:00.590 --> 14:01.990 The symbols are. 14:02.430 --> 14:08.910 This in logic and fs were like x or a line above the variable. 14:09.630 --> 14:11.350 Now the rule is. 14:11.510 --> 14:15.310 The rule is basically output the opposite of the input. 14:15.430 --> 14:20.670 If A is one and not A is zero and vice versa. 14:20.710 --> 14:26.630 So basically how we can write that is we will just have two. 14:27.590 --> 14:31.950 So we have a here and we have not a. 14:33.030 --> 14:34.190 This is so easy here. 14:35.030 --> 14:38.590 So if A is zero not A will be one. 14:38.590 --> 14:42.270 And if A is one then not A will be zero. 14:42.310 --> 14:42.910 That's it. 14:43.150 --> 14:48.430 The Not operator is like a mirror that always shows the opposite of what it sees. 14:48.430 --> 14:51.550 If the input is true, the output is false. 14:51.550 --> 14:54.510 If the input is false, the output is true. 14:54.750 --> 15:02.800 The reversal process is known as inversion, and it's why the Not gate is also called an inverter. 15:03.240 --> 15:05.360 Now let's use a practical analogy. 15:05.400 --> 15:08.120 Imagine you are looking at a room with a smart light bulb. 15:08.320 --> 15:11.800 The switch on the wall is programmed with a not logic. 15:12.040 --> 15:15.880 If the switch detects a signal off, it turns the light on. 15:16.040 --> 15:20.360 If the switch detects the signal on, it turns the light off. 15:20.600 --> 15:26.840 Now, this behavior is especially useful in logic circuits when you need to deny or cancel a condition. 15:26.840 --> 15:33.200 For example, a security system might activate an alarm only when a sensor signal is not triggered. 15:33.720 --> 15:34.840 So in programming. 15:35.080 --> 15:41.600 In many programming languages, the Not operation is written using this ternary symbol. 15:42.080 --> 15:52.640 So um, basically not true equals false, and not false equals true. 15:53.360 --> 15:54.930 And in Circuits. 15:54.930 --> 16:00.970 The Not gate is drawn as a triangle like this. 16:01.370 --> 16:13.930 For example a or let's actually represent it by x here triangle circle circle and what not x. 16:14.730 --> 16:19.570 Now that is how you draw in a logic circuits. 16:20.170 --> 16:27.050 The triangle represents the flow of logic and the circle means invert the input. 16:27.570 --> 16:29.970 And now you will encounter not gates. 16:29.970 --> 16:36.050 Frequently, when designing logic functions that require cancellation, inversion, or toggling a state 16:36.050 --> 16:38.290 from on to off. 16:38.970 --> 16:45.330 Now George Boole used multiple action for and and addition for or. 16:45.530 --> 16:46.010 Why? 16:46.290 --> 16:52.490 Because their behavior is similar to arithmetic, and using symbols made the system easier to work with. 16:52.730 --> 16:53.210 This idea. 16:53.210 --> 16:55.110 Connected logic to mathematics. 16:55.150 --> 16:59.430 Allowing logic statements to be multiplied like equations. 17:00.030 --> 17:05.710 Now that you know the basic logic operations, you will soon learn how to combine them to build more 17:05.710 --> 17:15.910 advanced logic gate like with X or which is exclusive or true only when one input is true, not both. 17:16.430 --> 17:24.790 You will learn that in next lectures and we have also Nand or non nor. 17:25.870 --> 17:28.350 These are the combinations of basic operators. 17:28.390 --> 17:34.510 Now you will also learn how these gates are wired together to create a circuit that can add, subtract, 17:34.510 --> 17:36.830 compare and store information. 17:36.870 --> 17:39.750 Now the way things your computer does every second here. 17:40.990 --> 17:45.350 Take your time to review the truth tables and practice drawing the gate symbols. 17:45.710 --> 17:51.390 This will help you build a strong foundation as we go further into digital logic. 17:51.630 --> 17:52.590 Thank you for watching. 17:52.630 --> 17:54.790 I'm Typhoon and I'm waiting you in the next lecture.