Transcripts
1. Introduction: welcome to this course in the schools. We're going to learn about logic Gates and Boolean algebra. These are some of the most fundamental topics in electron ICS and computer science. These topics will help you to learn the difficult concepts in electrons and computer sends in future. In the schools, I have explained topics with help off video lectures, which is supplemented by examples and quizzes. After completing the schools, you will be able to solve the difficult problems off bullion, algebra and logic. Kids with ease the schools were express for at once high school students and engineer insurance. However, anyone interested in these topics can take the schools, Thank you and welcome to the schools.
2. NOT Gate: in the schools. We're going to learn about logic Gates and Boolean algebra. So let's first start with logic Utes before we don't deep into logic Gates, let's understand the significance off logic Gates that Why do we need to learn about NASA kits in the first place? Actually, large includes our basic building blocks off all digital circuits. For example, your cellphone has billions and millions off large tickets that function together to make all the basic operations in yourself when possible. Now you can imagine how important it is to learn about logic. It's actually lodging. Dudes are also used in basic concepts off computer signs, so it becomes even more important to learn about our tickets. So lets for start learning about Lhasa. Kate with not gate. This right here is a symbol for not Kate. This circle symbolizes the not gate. This box over here, this signifies the truth stable. The truth table is actually the representation off input and output off a gate. Her E is the input, and why is the output had it is written? Why is equal to a bar? This is a bar is also called a compliment, and this compliment symbol signifies not are not gate. So the Stroot table, the input and output are represented in different columns. For example, in the first room, Zito represents the input. No, let's see. Zero is the input overhead. This is the input terminal for not get and this is the output dinner. So you represents the input. So the output that would be one. So I can write one over. Similarly, when the input this one, the output is equal to zero and large tickets. We're only concerned with binary numbers numbers only one and CTO and northern number. What not get does not gate reverses the input. It does one and two CDO and it can words Zito 21 percent Lee. When one is the input, it gives zero as output and when zero is then put gives oneness output. So that's what not get us an Each and every gate in the CDs is only concerned with Van Arena mus that its numbers one and CTO and one thing that is important to remember that this bars him but our compliments symbol signifies not gate and this circle over here this is also important. So that's all about not gate
3. OR Gate & AND Gate: now that we have learned about not gate, it's stamp for us to learn about our get. Now our gate has to input terminals that are represented by E and B and the output and is represented by why now, as the names scientist are for the output to be equal to one, neither e r B. That means either off. This input must be equal to one. So as you can see, we get one. Only Vin, either a. R B is equal to one. Are both of them are equal to one. Either are when both A and B are equal to zero. We get the dough here and for rest. All other cases we get one. You can compare this case with the kids off a test in which a team off to parties beats, Let's say, a team off A and B, and in this test, Zito zero represents failure, and one represents passing this test for passing this test. You need only one off this member off the steam to pass the test. If both of them can pass, then also, you'll pass them test. But for passing, you need only one off the member to pass. As you can see here, only one input terminal should be one for us to get output as one. This is our arc it Now we'll move toe End it now And it also has two terminals which are represented by E and B and our two terminals is presented by why as well No are are get really presented it by a plus B but for on gate real prison eight by multiplication symbol our dart symbol. As you can see here, in order to get one as our output, we need boot e on B to be equal to one. So it justifies the name. And here for rest, all other cases the art put a zero. And where we have A and B equals one, we get output as one. You can compare this with an analogy, as in the previous case in this test, where the road represents failure and wonder presents. Passing this test, we need both the member off the team toe pass in order to pass the test. So when both A and B passes the test that Dean de, which comprises off E and B, that passes which is quite up was it to what we see in the previous what we saw in the previous case
4. NOR Gate & NAND Gate: in this video, we'll learn about nor gate as you can see her. The symbol of nor gate is very similar to the symbol off or gauge the same goat. No audition. But have you can know this one difference that this circle this circle looks familiar, right? We saw I'd and gives off, not gate. So it is just our gate and not get put together. We have first we have our gate and then we have not gate. They are just good together, side by side. So in case off not get we saw that it just reverses the input. And in this case, the output off our gate becomes input for this. Not get This is input for argued and this is output. Forget and this output becomes not gives input. So what does not get does that? It just reverses the output off target and you can see it from here. Hair for 00 Rehab one. What a and P equal to zero. We have output as one which is just that oppose it as we saw in the our gate. In case off our get poor e and B both equal to zero. We had Cedo and for all other keys is we had output as one. But in this case, it is just the oppose it. So we can assume that nor gate is just a pulls it off our gate, our nor compliment is equal to are. And that's all because of the presence off this art in the output terminals with just reverses the output off our to make it oppose it off our. So that's all about our target, Andi. Now we have no handed and this is just similar to a previous case. This is just oppose it off and get how you can see for 1st 3 We have output as one which is just oppose it off off, and it in case off and gate. We saw that the output off 1st 3 cases where zero and output for both A and B equal to one was one and had what we see that that output has been totally reversed by this not so mand . It's just that oppose it off. And because of the presence off this not get at its output terminal. So that's all about our nan. It
5. XOR Gate & XNOR Gate: And now we're going to learn about SAARC eight Sark. It is represented by this symbol over here That symbol look somewhat similar to the symbol off or gate. But also we have to input on a single output. And now what is interesting here that it started this oneness output Only when one offers input is equal to one. When both of the input are equal to one, we get zero here and when both of them are equal to zero again gets you. So it is equal to one when inputs are dissimilar. When one off the input is equal to one on other input is equal to zero only when in that case that we get one. So that's all about our SAARC it. And now we're going to learn about X narc it. This xar gate is just the reverse off. Sark it in case off target. What? We saw that 400 We had Cedo who are zero when we had one. And for 10 we had one again and at last for 11 We had see you. And what you can see is that this X nor gate is just the reverse off. Sark it. So what regarding it's our get. It is just the opposite of that. Just like nannies Up. Was it off? And so that's all about our X market. Now I have a problem for you. And our gate has three inputs. The value off one off the input terminal is equal to zero while the values off other too. I have one. What will be the output value? You can pause the video and try this problem yourself. After some time, I'm going to reveal the solution. Now the answer for this question. His boot. Why so because And our gate What we saw that even if one off the value is equal to one, we get one a doubt. So in this case, we had three inputs sound drawing on our gate with three inputs. Do off them had value zero and one off them and value when. So even if even if one of the input has really one, we get one as the output and in this case, we have to values is one. So we get one here
6. Basic Boolean Arithmetic: Now we're going to London, hook loss off bullion and TV before we dive deep into the loss of billionaires era. It's important for us to know basic binary arithmetic, so let's get started. So the first equation over here it says one plus zero is equal to one. Now we all know this from our basic mathematical knowledge that one plus zero is equal to one, but her It's quite different because in the second equation, what we see is one plus one is equal to one. Now in our normal mathematics, the mathematics that we use in our day to day life one plus one would be equal to two. But in binary mathematics, we can't have any other number except zero England. And the reason we get one hair. There are multiple ways to describe it, and I'm going to describe it. Using are different, but that this blessed sign over here the signifies our gate. What an R get It's input. Let's say a and be they are written as e bless be, and these places over here are nothing but these places for our get Now, as you know that for input one and Zito who are an r get we get output equal to one. So what the secretion says is that when the input for an R gate is one and zero, we get output as one and that we know from our knowledge off arc it now what disagrees In seizes that went input for an argument for boot. The input terminals are one and one, respectively. We get output as one, as you might know, that from your knowledge off logic gates that when the input for an R get is equal to one and one, the output is one. So this equation just sees that, and this equation is quite important. Also, it's quite counter into too. So it's important to remember this now. This is quite in to do that one multiplied by zero is equal to zero. But Herr also these dot symbols This multiplication symbols actually signifies ended. We know that went for an ang gate. The input are one and zero. We get zero as input and the symbol for and it is a dot So this daughter were here signifies, and what the secretion says is when input foreign and it is one and zero we get out, put a zero. But for simply for the case off simplicity, you can assume your experience from your normal man had access. One multiplied by zero is equal to do similarly have one multiplied by one is equal to one . But actually it is saying that when input for an and gate is one and one, respectively, in both the terminals, we get output as one, but you can assume it. That one multiplied one is equal to one for the sake of simplicity. The only aggression that that is quite odd over here is one plus one is equal to one, and our our gate say's Why's it so? So you might remember this equation for future reference. So that's all about our basic by Nadiya.
7. Annulment Rule: Now that we have learned about basic finally arithmetic, it's time for us to learn about an elemental, which is the first rule in our series off laws off Boolean algebra, so has the first rule. A plus one is equal to one. Her E is a bully and reusable. Now, unlike mathematical, variable bullion really able can take only two values. That is, it can either be one are again, be equal to Zito, and no other values are possible either. A can be one what it can be. Zero. So let's first test for values one one Bless one. Now from basic been area Atlantic, we know that one plus one is equal to one. So when is equal to one, we get one in the right hand side. So this aggression satisfies when is equal to zero. Zito bless one now zero plus one is also equal to one. So again this equation is satisfied. So that's how we prove the secretion. Similarly, for the second aggression, a multiplied by zero is equal to zero. He can again take only these two values. So when is equal to one one multiplied by zero R one and zero is equal to Zito. See, we're getting zero here in cedar here. So the situation is satisfied for equal to one. And for a equal do Zito substituting the secretion you get zero multiplied by Zito, which is again equal to zeal. So this equation is again said, despite for Zito. Also, since they can take only two values that are when and CD so that's all about her, an element truly.
8. Identity & Idempotent Rule: And now we're going to learn about identity that sees a Placido is equal to a which is quite into Redux now is a brilliant were Ubell here. What is who says is Ah bullion variable Bless of zero is equal to that 1,000,000,000 valuable and a brilliant very were towns. One is also equal to that Boolean variable. No, there's nothing special about this. A I can write b plus zero is equal to be are be Diane's one equal to will be. So if I take ah equals one, it will give me one plus Zito equal to one. This is true, of course. And if I take equals zero, it will give me zero Plus Zito is equal to has a Zito zero So that proves that this equation is true Similarly if is one here one dines one a is one So again the secretion is true one tons If is Ito CDO turns one is equal to e which is equal to zero which is true one second So since they can take wanted Thies to value that one and zero we don't need any other ready So that's all about our identity toe now we have ID important and true, which sees ah bullion valuable Plus another 1,000,000,000 reliable is equal toe that 1,000,000,000 with no, I'm gonna prove how. Let's see the Boolean variable has value one that means is equalled one now substituting in the situation one plus one We know that one plus one is equal to one and that satisfies the secretion. We know that value is equal to one. So the situation and satisfy Nazi is equal to Zito No aceto less Zito equals zero. Which is true, of course. So that Bruce this equation now there's nothing special about this. E As I've said in previous kids, I can say that BC this BC he's equal to bc These are also bullion medieval just like so. This is true for any 1,000,000 video. Similarly, using this process, you can prove this equation Also, this is letters access for the student. So that's all about Oh, I'd important. True
9. Complement & Double Negation Rule: Now we're going to learn about Comitatus Law, which is pretty simple. What this law seizes a plus B is equal to B B plus. Putting it simply one plus zero is equal to zero place when and for this one. One times Zito is equal to zero times. One that's on this lot sees pretty simple. Now we're going to learn about double negation. What this rule seizes was, we need to consider this e bar this e but is nothing. But this is just in worse. Off this is called a compliment. The symbol we have already seen. The symbol in case off a not gauge or not get at the input is a The output is a compliment . So when input zero output is one, when input is one output, it's Zito. So that's what this compliment is not see. When e is equal, do one e body is equal to zero, and then a is equal to Zito E. But equal to one are a complement is equal to one. So compliment is just the opposite off a valuable if the variable zero. Its complement is one that the variable is one. It's complimented. Zero so when a variable has double negation or it has double competent, as in this case, say reasonably it waas equal do one. So for fresh negation a became zero a compliment is equal to zero and for a double compliment are a double negation. This is the complement off a compliment, which is a compliment off zero. So it is one second one. So what we see is a double compliment is equal to a which proves that this human as correct a double compliment as he quit, too. Hey, that's what we get from here. So that's that song for double negation room.
10. de Morgan's Theorem: Now we're going to learn about compliment law. Basically, Compliment Law says that Ah bullion, Really? But that's the compliment. Off that bullion variable is equal to one. Similarly, a bullion video dance. The compliment off that Julian really able is equal to zero. Now let's try to prove them who has less proved this one. Let's take equals one. Now they can take only too well use either one a zero. I'm I'll be proving this serum for E equals one. So self shooting equals one into the situation. It becomes one. What a hair bless a compliment. It is equal to one. So one compliment zero So right zero. Now I know that one plus zero is equal to one So that crews this period Similarly, for eight ends e compliment equal to zero mystic equals zero. No. So stooping this into the secretion zero dinz a compliment since is equal to zero. A compliment will be one. So one dance Edo that will be equal to Zito. Now you can prove this for equal to one. So that's all about our compliment law. Basically, it says that a brilliant video, plus a complement of that 1,000,000,000 variable is equal to one and a Boolean variable dance. The complement of that William Variable equal to zero. Now that's more to DeMarcus. Now. What D Morgan students sees is a plus B compliment is equal to a compliment. Dansby compliment Now one thing that should be kept clear from the big name that e Bless Be Compliment is not equal. I repeat, it is not equal to a compliment, plus b compliment. This is not necessarily equal to e compliment plus b compliment. So never assume that this is equal to a compliment, plus b compliment. So what the students says is that this compliment off some off different, really invariable is equal to the product. Off Compliment off those brilliant, valuable putting it simply, let's say, is equal to one and B is equal to zero. So e bless Zito compliment. Using substituting in this equation a place zero compliment. We know that one plus zero is equal to one, so it will be one compliment, which will be equal to zero. No, in the artist side, I mean this side e, but plus Bieber is equal to one and bees equal to zero. So that will be zero times a the lieutenant's one. So that will be again equal to zero. So that bruise this, that very presidents actually, the proof off this serum a spit involved and it needs some higher mathematical knowledge. So I'm going to skip this. If you're interested, you can look it. Look into it. So that's all about Dean organs here. Similarly, if ah, a plus B compliment is equal to e compliment plus B compliment, that means it's segregates is really able and this Crowder genes into some, and we righted this week. These dudes are quite important. So it is advised that you should remember this, so that's all.
11. A strategy for solving problems: Now I'm going to discuss a strategy for solving problems involving bullion and era. So has the Step one reduce the individual terms into their simplest form. Hair. I have a problem that I used to discuss this stategy. So Step one is to reduce the individual terms. So what I see here that I have to individual down here, this one and a is already in its most reduced form. So what I need to do is I need to reduce this dome over hair into its simplest form. So Harry, go using De Morgan's to him. What? I went to get it E compliment and be Since there is already a compliment over here, I'm going to get be double compliment. This aerial simply mean a This place will same dreaming. Plus, now step does is take the common terms. But what I see here that there are no common terms over here. Hungering dues had forward to step three with the simplify very meaning expression, so simplifying and what I see that b has a double compliment. So it's going to become be just be. Since beatable compliment is equal to be, it's going to become be Andhra Meaning dancer? Yeah, e compliment. Bless. Hey, no, A compliment. Plus a it is from the compliment rule. You know that a compliment less is equal to one, So this will become one. So here I get be Bless what? From an elementary rule? What we know is one plus anything. Any Willian really able Plus one is equal to one. So finally we get one away her answers one No, I have a problem for you. This one You can pause the video to solve this yourself. And I'm going to discuss this solution after sometime. So now I'm going to discuss the answer. The answer is one. It will be a bar. So going forward with our sanity. If we apply you feet, simplify the individual domes. Plus we need to simply fight this one. So, using De Morgan's law, it will be e bar dinz. Be double complement our double bar. Bless you. Compliment. Now, the second step was taking comment Dems. Now what as he is a compliment is going over here, so I'll take e compliment as Komen. Now we also know that from double negation rule that be double compliment is equal toe nothing but be only so right. Be here now since we had taken a compliment is common. This will become one now from an elemental. We know that anything plus one is nothing but one only. So it will be a compliment. Times one which will be nothing but e compliment. So that's something. So that's why our and said it's one. It is a compliment. Now I have another problem for you, which is also an important property. This is a place. Beaton's B Plus C is equal to a plus. Beat and see. No, this is left as excites for the student. This is very easy. All you need to do is you need to expand this. I need to apply the basic properties off Boolean algebra off basic rules, and you're going to solve this. So that's all for this video.
12. Example-1: Now we're going to solve setting examples on a Boolean algebra. So this is our first example, which says simply by the Given Boolean expression and the expression is a B compliment plus a B so asked me to write it down over here, you big compliment, plus a B. Now the first stuff for solving a 1,000,000,000 every problem is to simplify the individual terms. Now these stunts can be in and simplify any further. So I will move two seconds up, which is to take the common domes. What I see that is comin over here I'll take a as a common toe, which leaves millet e Dunn's Be Compliment Plus B Now from compliments rule. I know that we know that Be compliment plus B this equal to one. So this is nothing but eight domes who won, which is nothing but a So this problem is pretty simple. This is just too explain. The fundament does so at us. We get here. So that's all
13. Example-2: Now we'll solve one more example on bullion. Debra, this is example Number two express the given bullion expression and simplest form. Now, when I look at this, what I see is the first time is not in its most reduced from remaining Do Dems are already in in their most reduce homes Onley system and still be reduced further. So applying De Morgan slow I can write This is a job plus be compliment less nurse Be see Yeah no, we get this. Come. This is what they're using DeMarcus no from double navies in true, This will be equal to you So it will be e miss Hey, Must be compliment plus BC Now this is a place a will be nothing but even from Adam Portent route now unranked bc BC Now, it's time for us to take the comments which is step doing her status. So I'll take years common between these two terms This one this So taking is common This will become one This will become being see now one plus B c from annulment rule We know that one presenting an equally invariable is nothing but one so dry one eight ends one over here because this becomes one one plus any 1,000,000,000 review . But this one only bless B company. So finally, what we get us? Yeah, Miss, be complement. This is our final expression and it can be reduced. One thing that I want to see is that our started C is not absolute or standard one. You can use any other strategy. I have devised a strategy to make this process easier, but it's not absolute stategy. You can use any understand egy also, if you feel so but this status easy that is reducing and taking common, it's quite easy to do so I have says a stiffness, so that's all about 26
14. Example-3: Now we're going to solve one more example on William and Deborah, This example says, expressed the Given William expression in its simplest form. Now, in the given expression, this only the second dumb needs to reduce further. According to our stategy rest, individual dumps are already in the most read useful. So let's get started now. E. Come from NBC will simply remain as a compliment. BC bless blang de Morgan's to him. It will. It will be a compliment. Dansby See complement this place. Welcome convert. In two times it will be a compliment. Dimes B sico Applying de more gust here. Bless, See e condiment. This will mean as it is now, this BC compliment needs to be reduced for So they come from NBC will remain as cities a compliment. B c will remain as it is. Bless eight times this BC compliment will convert into be compliment plus c compliment. Applying deem organs theater. It just says that BC compliment is equal to be compliment. Less see, compliment plus see a compliment and feel again dreaming as it is. No, I need to explain this further. So a compliment. Well, you see Yes. A compliment be compliment. This is not a big compliment. This is a compliment. Be compliment, expanding from this plus e compliment. See Congress? Let's see. Contract Now Our second step is to take comment terms. Now what we see here that a compliment is common in all terms, as you can see. So I can't take a compliment as common term over all the individual doors. So it will give me BC bliss being compliment. Unless seeing compliment yes, see now the sea compliment, let's see, will be equal to one from Complement Room. The third step was to simply pride further, so we're simply trying it for the so again writing the compliment BC Yes, be compliment. See compliment plus C, which is equal to one No, the bullion really able when we get blessed. One. We should ignore everything except one we knew from. And I am a true that William very well plus form is our physical +21 So whenever you get anything plus one, just ignore that and write one over there so it will be equal to a compliment. Bang swim, which is nothing But Yvonne drew it so simplifying this expression. Finally, what we get is e compliment. So that's our answer