>>Good day. This is Jim Pytel from Columbia

Gorge Community College. This is Digital Electronics One. This lecture is entitled

Universal Properties of NAND and NOR Gates. We have discussed a bunch

of basic logic gates. We first discuss the

AND, the OR and the NOT and then we introduce

the NAND and the NOR and their negative logic

equivalents to negative AND and the negative OR. And then we also introduce

the EXOR now that’s exclusive OR exclusive NOR gate. I showed you how you can

make all of those other gates with combinations of the

AND and the OR and the NOT. However, what if I was tell you

it’s not really done that way. You use another basic

component to make the ANDs and the ORs and the NOTs. And if that’s true, if that one

basic component is making those basic gates it sounds like

what I’m trying to tell you is that one basic component hooked up in different configurations

is making any logic circuit that we can possibly create. What it really is you’re

actually making is these NAND and NOR gates. What’s inside a NAND

and a NOR gate? Well for this course

you don’t need to know. For some of our connective

devices and circuits you’re going

to crack these things open. You’re going to make

one of these things. It’s super simple

what’s inside that. That’s why you use the

NANDs and the NOR gates. They’re super simple to make. They’re super simple

to mass produce. And if you can hook them up in different configurations

you can create a NOT, an AND and an OR. You can use a NOT, AND and OR. You can make a NAND. Well you’ll see it’s easier. I would say how do

you make an AND out of a NAND, you just use a NAND. You don’t have to do

anything funky with that. A NOR, a negative OR a negative

AND exclusive OR, exclusive NOR and you take all of these

things and crazy combinations and build higher functional

combinational logic circuits. And sequential for that this

lecture is going to be two easy. It’s an incredibly important

topic but it is too easy to understand what I’m

going to talk about here because you guys are experts

by now and you can go ahead and spit out the truth table

for a NAND and for a NOR without looking at

your cheat sheet. And if it took you longer

to write that truth table on your note pad, because you’re

actively listening to this thing than it took me to copy

and paste it, then you need to go back and look at

the NAND and the NOR gate. If you’re still looking

this thing up right now you

got some work to do. You need to know how

these things work. When I say no, you

need to be able to at a moment’s notice spit out

the truth table for these things without consulting

some resource. That being said, I’m going

to leave the NAND gate and the NOR gate’s truth table

here for those individuals that potentially need some

assistance we may be even consultingly six. So, the universal properties of

the NAND and NOR gates essential to this is obviously you

understanding the NAND and the NOR gates and

how they work initially. But I’ve already referred

to this in the NAND and NOR gate lectures. This guy how do I make a NOT

gate using just a NAND gate. Let’s just stick to

the NANDs right now. How do I make a NOT gate? Alright this is take a NAND. It’s got two inputs. Tie its inputs together and

put a digital data set on it, up down up down up down. What are the two inputs getting? It’s getting the same value. So when it’s zero, when that

waveform A is zero what are the two inputs getting? Zero and a zero. Look at the truth table. It’s getting the

zero and a zero. What’s my output? A one. Oh my gosh

it’s inverted it. When that digital data is

getting a one what are my two inputs getting? A one and one where

on my truth table? If a one and a one are on

the inputs what’s my output? Zero. Oh my gosh,

it’s an inversion. So if a NAND gate with its

two inputs tied together it is functionally equivalent

to a NOT gate. And if you can understand that

you can build an AND gate. And if you can understand

that you can build an OR gate. Okay, I’m challenging you

to go ahead and do that. If this is true right here,

if I can swap out a NOT gate for a NAND gate with its two

inputs tied together how can I create an AND gate

of just NAND gates? And I need you to

pause the lecture. I need you think about this. How do you do it? If you pause the lecture

you’d probably come up with the following action

given the NAND gate really is functionally equivalent

to an AND gate with its output inverted, right. So it’s A, B what is this? It’s A AND B. Well

it’s A AND B inverted. You may be using rule

nine which states that A, something inverted,

inverted again is equal to A. Why not put

another inverter here? So if this thing is a NAND

gate and I can make an inverter out of another NAND gate

what’s that look like? It looks like this, NAND

gate inputs A AND B output into another NAND gate with

its inputs tied together. That’s your AND gate, functionally equivalent

to an AND gate. Don’t believe me, put

it in the truth table. AB zero zero, zero

one, one zero, one one. What’s the output

of the NAND gate? A AND B inverted what

is the truth table? Don’t even look it up. One one, one zero. The only time it’s low is

when all inputs are high. What is this guy? Well it’s getting one and a

one for this first thing here. When it’s getting

a one and a one where are we on this

truth table? A zero, a zero. Now we’re getting over here

both inputs are getting a zero. What’s their output? Output is a one. What’s that truth

table look like? Well it’s A AND B

inverted, inverted again. What’s double inversion? A AND B. What’s that

truth table look like? It looks like an AND gate. Okay this is really cool and what you do is

just use this thing. Think of this thing as a box. And here is a box with inputs

A and B on the left hand side. I’ll put X on the right and

I know there’s a NAND gate in there feeding

under the NAND gate with its two inputs

tied together. But what’s it performing? So same thing as

what’s inside this box. It’s got two inputs

A AND B on the left and an output X on the right. And it’s performing that same

truth table as an AND gate. Let’s just take it a

little step further here. Okay, we’ve talked about using

a NAND gate as an inverter. Now we talked using

the NAND gate in different combinations

to make AND gate. This is a little bit tricky. How do I make an OR

gate out of a NAND gate? So look at our little symbols

here which we’ve drawn. I want you to find the

negative logic equivalent of the NAND gate. The negative logic equipment for a NAND gate is this guy

right there, the negative OR. See if you can take a negative

OR which is NAND gate and see if you can make a, excuse

me an OR gate out of that. Hopefully your drawing

looks like this. Okay so there is my negative OR. I use an OR with

its inputs inverted. How do I get rid of

that inverted input, inverted again right there? Well I just told you that

I could take a NAND gate and tie its inputs together

to make an inverter. What am I creating here? Because a negative OR

is equivalent to a NAND. I’m creating a functional

equivalent of an OR gate inside a box ABX. I look at this from

the box inputs A AND B of an output X that’s

functionally equivalent to an OR gate inside a box. Don’t believe me? Go ahead and try it out. So I got A. I got B. What

is happening to A AND B? I’m inverting them. NOT A, NOT B, pen issues. What’s happening to NOT

A and NOT B. So NOT A, NOT B. I’m ANDing them together

and then inverting the result. I’m basically doing

the NAND of this. The output is low when all

inputs are high right there. What’s that truth

table look like? That looks like OR

gates truth table. The output is high

when any input is high. And I’m saying this is

the input right here because that’s what I

originally start with. Don’t think of this as

the input because we’re on this side of the box, okay. That is how you make AND gates. That’s how you make NOT

gates, AND gates and OR gates out of just the NAND okay. And there’s something you might

want to put on you cheat sheet but man it is so easy. It’s so easy. You don’t even need to put

that on your cheat sheet because you already know

what I’m talking about. A NAND with its imputs tied

together that’s a NOT A. A NAND feeding a NAND based

inverter, that’s an AND. A NAND with its inputs inverted

with a NAND based inverter, that’s the functional

equivalent to a OR gate. So I just showed you how to do that with NAND gates,

not challenge it. Figure out how you do

that with NOR gates. And I’m going to throw a bone

to you what is a NOR gate with its inputs tied together? If you can understand

this it should be too easy to create a NOT gate. It should be too easy

to create an OR gate. It should be too easy

to create a NAND gate. Okay if you’ve done

what I’m asking you to do what is a NOR gate with

its two inputs tied together? Let me clean up because I was

talking about NANDs to that. Okay there you go. If I’ve got a digital data

signal coming in up down up down up down it’s

taken the value zero. What are the two inputs getting? It’s getting a zero and a zero. What’s my output? One, okay. I’ve inverted it. Now it’s a one. Both inputs are getting a one. My output is zero. I’ve inverted a NOR with

its inputs tied together. Functionally equivalent

to an inverter. How do I make an

OR out of a NOR? Well, it a NOR is an inverted OR you’re basically ORing

something then inverting it, why not reinvert the output

using a, you guessed it, a NOR with its inputs tied

together functionally equivalent to an OR gate. If a NOR is equivalent

to a negative AND, it’s negative logic

equivalent, how do I make an AND out of just NOR gates? Well take the negative logic

equivalent, a negative AND and invert the inputs. But I want to use just NORs. Take a NOR and tie

its inputs together. Take a NOR and tie

its inputs together. And since I want to use

this same symbol it’s going to look like this. And there’s the whole

cheat sheet. You’ve got your NOR with

its inputs tied together. It’s functionally

equivalent to a NOT. Got a NOR with its

output being inverted with a NOR based inverter

that’s equivalent to an OR. And then we’ve got an

AND gate which is a NOR with its inputs inverted

using NOR based inverters. So this is really neat. You’ve got many different

logic functions all formed from this same gate. This is, everybody in your squad

is shooting 5.56 ammunition. No one’s got some crazy

weird caliber musket ball that you can’t use

in your rifle. It’s the same piece of

equipment used over and over in different combinations so you can perform crazy

different logic functions. Hopefully this is easy for you. This should be, it should have

been a pretty easy discussion. No use beating a dead horse. Let’s just see if we

can apply this thing. Just take a logic circuit

and see if you can make that same logic circuit using

just NANDs or just NORs. Here’s an example right here. Okay here’s our, put our cheat

sheet off to the left hand side. I don’t know if you need it

but let’s go ahead and see if you can for the

output X right there, for that particular

logic circuit for a three input logic circuit. Go ahead and see if you can

create a NAND equivalent. What I’m going to do to do this

is draw a box around the OR, draw a box around the AND. Take out the contents of the box

and put in the NAND equivalent of what was in the box. It should look like this. It’ll make our life ever so slightly more easy,

make that a red box. There you go. Too easy to do the exact

same thing, try to implement that circuit using

just NOR gates. You should look at something

like this and there you go. That is the NOR equivalent of

the original circuit we started with and that’s kind of, that’s

what I would call correctish. It’s correctish. Yes, it’s performing the

output X like we’d expect and I dare you guys

to do the truth tables for the original one,

for the NAND based logic and then for the NORs. It’s going to come up

with the same truth table and the expression used. These are great,

great opportunities to do some De Morgans’

theorems and Boolean and logic simplification. However, there exists

something in this bottom circuit at the NOR based circuit that

I could potentially simplify. And when I say make this

circuit using just NOR gates, I need to actually kind of insert a kind of

qualifier to that. Use a minimum number

of NOR gates. Is there something

in our lower circuit that we could potentially

save some connections? And what I’m referring

to is right here. What is this guy doing? Well it’s a NOR gate with

its inputs tied together. What’s that? It’s a NOT gate. What’s this guy? It’s a NOR gate with its

inputs tied together. What is it doing? It’s acting like an inverter. Why in the heck’s name

am I taking a signal, inverting it, uninverting it. Okay, what am I using there? I’m using rule nine,

the double inversion. Sounds like it can

save some connections, forget those two NOR gates. What does my simplified

circuit look like? It looks like this. There you go. Problem with that though is

I can’t really define them in the red boxes and the

blue boxes like I did before. If I had purple, one of

my least favorite colors, what would I have to do is

just do an overlap right there. If I required that intermediary

signal I’ve kind of lost it in this implementation. Go ahead and see if you can

come up with, see if I said that these are great

opportunities for multilevel circuit analysis

and De Morgan’s theorem. Try to do the De

Morgan’s theorem. [inaudible] with some

Boolean simplification. Same thing for that. You should come up with the

same results and this one too. Try one more example of this and

then just put this topic to bed. I mean this is too easy. Just take the box, take the

contents out of the box and put in the box the equivalents. Try one more. Okay so here’s another

example that’s like a four input

circuit this time. And I’m going to give you

a challenge on this one. You can, yeah you

should instantly go for yeah, that’s easy. But I’m going to give you

a challenge this time. You can only use two

input NAND gates. Okay, so try to input this

circuit using just two input NAND gates. Try to implement the circuit

with just two input NOR gates. Okay, so hopefully you’ve come

up with something like this. Well first off, how do

I make this circuit? I’ve got three input AND gate. Don’t worry about the

NAN conversion right now. How do I make a three

input AND gate out of just two input AND gates. It should look something

like this. Okay, so what is

coming out of here? A ANDed with NOT B.

What’s come out of here? C OR D. What is this

last AND gate doing? It’s ANDing A AND B with C OR D. That’s the same expression

that’s happening here. It’s taking A ANDing

with B. Excuse me, ANDing with the results

of C OR D. Same expression and what I’ve done is I’ve

created a three input AND gate out of two nested AND gates

using the associative property of AND gates, okay. So that might help

out a little bit. Go ahead. See if you can come

up with a NAND equivalent. Hopefully it should look

something like this. And there you go, there’s

the NAND equivalent. There are opportunities

for some, actually let me just draw

this here quick for you. Where is the inverter for B? It’s that one or it’s this

AND gate it’s that one. OR gate it’s this one. Basically just taking

the contents of the box and just putting in the

NAND’s equivalent of it. Is there opportunities

for simplification? No. Can I redraw

this thing perhaps in a different fashion, maybe. You know maybe I could use like the negative logic

equivalent, the negative OR. It’s not really going

to try to help us. I know seems more complicated, but if the NAND is an

incredibly cheap part to make, it’s pretty neat that you can go

ahead and take and think of it in terms of a PLD,

programmable logic device. Here is just a sea, an

endless sea of unhooked up NAND gates stretching

out to infinity. And if I could just somehow

program these unconnected demands to perform NOT gates

and perform AND gates to perform OR gates and then take

those NAND based NOTs, ANDs and ORs to create NANDs. Well I would say you’re

not going to create a NAND out of it because it’s a NAND. NOR is exclusives ORs,

exclusive NORs and then build up to make adders

and it’s pretty cool. You know it’s a neat

way of doing things. Okay, here’s another challenge. Take the same circuit, make

a NOR based equivalent of it. Okay there is my initial attempt at a NOR based equivalent

of that. Where is the inverter for B? Right there. Where’s that first AND gate? Whoops got to change

the color, right there. Where is the OR that one? And where is my final

AND gate right there? Where in here might I be able

to simplify, reduce the number of connections that I need? And long story short it’s

always those double negations when you start looking at these

things, start looking for times where there’s an inverter,

a NOR based inverter or a NAND based inverter,

because you can do this for NANDs where they are two

inverters feeding each other. And why would you ever

do something like that? Okay so think about right

here and right there. I can remove them entirely. I don’t even need

those things, okay. Let’s go ahead and

redraw it again. So there is the same circuit

using the minimum number of NOR gates. What have I lost? I’ve lost access to

that C OR D signal. And additionally I’ve lost

access to that NOT B signal. Potentially you know

you need NOT B and C OR D for some other

logical output. You know here’s X. Here’s Y.

You may need those things. What have you gained here? Well if you don’t need those

things don’t worry about it. What have you gained? Less power consumption. If you really are

implementing these things out of fixed function of your

eight circuits, less gates. That’s it guys. I mean this is really,

really simple. You should the know the

truth table of the NAND, negative logic equipment,

the negative OR and the NOR and it’s native logical

equivalent, the negative AND. And for you to understand

this you should be able to use again Boolean rules to go

ahead and simplify this thing. If you’ve got those double

inverters feeding off each other, why would you ever

do something like that? Okay, let’s bring this

lecture to a close and move onto the next subject.

## 16 Comments

## victor kh

Thank you so much! Incredibly interesting yo learn it this way

## Ali Raza khan

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## Kaim Jutt

thnx

## Mahamud Hussein

oh my gosh!

## Patriot2020

pretty neat! thumbs up

## Keshari Govind

Thanks. Very helpful

## Goodluck Gabriel

Amazingly well explained, thank you alot sir

## Vidhur Savyasachin

perfecto

## Ramachandruni Chandana

video is simply astonishing! thanks allllot sir for this wonderfull vedio!

## TheOrbitalDropShock

A(B+C) = (AB)+(AC)

Then shouldn't I be able to make the example at 12:08 with just 3 2-in-NANDS?

A NAND B and A NAND C into another NAND (inverting after first NAND(s) to get AND(s), and inverting before second NAND to get OR cancel eachother out)

## Hyder Ali

you are awesome, you are beautiful, you are sexy

## Shiva Kunwar

no offence, but I felt it lot more complicated with this video..

## supernovaification

Really great explanation. You were clear, concise. Everything I look for in a quality lecture. Look forward to future content!

## Abdulrahman Abomazid

You are legend !

😉

## Fardad Ansari

Thanks for showing us how can we think of logic simpler that previous ones we`ve covered

## Mashable

CLEAR AND WOW