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• Hi, I’m Carrie Anne and welcome to Crash Course Computer Science!

• Today we start our journey up the ladder of abstraction, where we leave behind the simplicity

• of being able to see every switch and gear, but gain the ability to assemble increasingly

• complex systems.

• INTRO

• Last episode, we talked about how computers evolved from electromechanical devices, that

• often had decimal representations of numberslike those represented by teeth on a gear

• to electronic computers with transistors that can turn the flow of electricity on or off.

• And fortunately, even with just two states of electricity, we can represent important information.

• We call this representation Binary -- which literally meansof two states”, in the

• same way a bicycle has two wheels or a biped has two legs.

• You might think two states isn’t a lot to work with, and you’d be right!

• But, it’s exactly what you need for representing the valuestrueandfalse”.

• In computers, anonstate, when electricity is flowing, represents true.

• The off state, no electricity flowing, represents false.

• We can also write binary as 1’s and 0’s instead of true’s and false’s – they

• are just different expressions of the same signalbut well talk more about that in the next episode.

• Now it is actually possible to use transistors for more than just turning electrical current

• on and off, and to allow for different levels of current.

• Some early electronic computers were ternary, that's three states, and even quinary, using 5 states.

• The problem is, the more intermediate states there are, the harder it is to keep them all

• seperate -- if your smartphone battery starts running low or there’s electrical noise

• because someone's running a microwave nearby, the signals can get mixed up... and this problem

• only gets worse with transistors changing states millions of times per second!

• So, placing two signals as far apart as possible - using juston and off’ - gives us the

• most distinct signal to minimize these issues.

• Another reason computers use binary is that an entire branch of mathematics already existed

• that dealt exclusively with true and false values.

• And it had figured out all of the necessary rules and operations for manipulating them.

• It's called Boolean Algebra!

• George Boole, from which Boolean Algebra later got its name, was a self-taught English mathematician in the 1800s.

• He was interested in representing logical statements that wentunder, over, and beyond

• Aristotle’s approach to logic, which was, unsurprisingly, grounded in philosophy.

• Boole’s approach allowed truth to be systematically and formally proven, through logic equations

• which he introduced in his first book, “The Mathematical Analysis of Logicin 1847.

• Inregularalgebra -- the type you probably learned in high school -- the values of variables

• are numbers, and operations on those numbers are things like addition and multiplication.

• But in Boolean Algebra, the values of variables are true and false, and the operations are logical.

• There are three fundamental operations in Boolean Algebra: a NOT, an AND, and an OR operation.

• And these operations turn out to be really useful so were going to look at them individually.

• A NOT takes a single boolean value, either true or false, and negates it.

• It flips true to false, and false to true.

• We can write out a little logic table that shows the original value under Input, and

• the outcome after applying the operation under Output.

• Now here’s the cool part -- we can easily build boolean logic out of transistors.

• As we discussed last episode, transistors are really just little electrically controlled switches.

• They have three wires: two electrodes and one control wire.

• When you apply electricity to the control wire, it lets current flow through from one

• electrode, through the transistor, to the other electrode.

• This is a lot like a spigot on a pipe -- open the tap, water flows, close the tap, water shuts off.

• You can think of the control wire as an input, and the wire coming from the bottom electrode as the output.

• So with a single transistor, we have one input and one output.

• If we turn the input on, the output is also on because the current can flow through it.

• If we turn the input off, the output is also off and the current can no longer pass through.

• Or in boolean terms, when the input is true, the output is true.

• And when the input is false, the output is also false.

• Which again we can show on a logic table.

• This isn’t a very exciting circuit though because its not doing anything -- the input

• and output are the same.

• But, we can modify this circuit just a little bit to create a NOT.

• Instead of having the output wire at the end of the transistor, we can move it before.

• If we turn the input on, the transistor allows current to pass through it to theground”,

• and the output wire won’t receive that current - so it will be off.

• In our water metaphor grounding would be like if all the water in your house was flowing

• out of a huge hose so there wasn’t any water pressure left for your shower.

• So in this case if the input is on, output is off.

• When we turn off the transistor, though, current is prevented from flowing down it to the

• ground, so instead, current flows through the output wire.

• So the input will be off and the output will be on.

• And this matches our logic table for NOT, so congrats, we just built a circuit that computes NOT!

• We call them NOT gates - we call them gates because theyre controlling the path of our current.

• The AND Boolean operation takes two inputs, but still has a single output.

• In this case the output is only true if both inputs are true.

• Think about it like telling the truth.

• Youre only being completely honest if you don’t lie even a little.

• For example, let’s take the statement, “My name is Carrie Anne AND I’m wearing a blue dress".

• Both of those facts are true, so the whole statement is true.

• But if I said, “My name is Carrie Anne AND I’m wearing pantsthat would be false,

• because I’m not wearing pants.

• Or trousers.

• If youre in England.

• The Carrie Anne part is true, but a true AND a false, is still false.

• If I were to reverse that statement it would still obviously be false, and if I were to

• tell you two complete lies that is also false, and again we can write all of these combinations

• out in a table.

• To build an AND gate, we need two transistors connected together so we have our two inputs

• and one output.

• If we turn on just transistor A, current won’t flow because the current is stopped by transistor B.

• Alternatively, if transistor B is on, but the transistor A is off,

• the same thing, the current can’t get through.

• Only if transistor A AND transistor B are on does the output wire have current.

• The last boolean operation is OR -- where only one input has to be true for the output to be true.

• For example, my name is Margaret Hamilton OR I’m wearing a blue dress.

• This is a true statement because although I’m not Margaret Hamilton unfortunately,

• I am wearing a blue dress, so the overall statement is true.

• An OR statement is also true if both facts are true.

• The only time an OR statement is false is if both inputs are false.

• Building an OR gate from transistors needs a few extra wires.

• Instead of having two transistors in series -- one after the other -- we have them in parallel.

• We run wires from the current source to both transistors.

• We use this little arc to note that the wires jump over one another and aren’t connected,

• even though they look like they cross.

• If both transistors are turned off, the current is prevented from flowing to the output,

• so the output is also off.

• Now, if we turn on just Transistor A, current can flow to the output.

• Same thing if transistor A is off, but Transistor B in on.

• Basically if A OR B is on, the output is also on.

• Also, if both transistors are on, the output is still on.

• Ok, now that weve got NOT, AND, and OR gates, and we can leave behind the constituent

• transistors and move up a layer of abstraction.

• The standard engineers use for these gates are a triangle with a dot for a NOT,

• a D for the AND, and a spaceship for the OR.

• Those aren’t the official names, but that's howI like to think of them.

• Representing them and thinking about them this way allows us to build even bigger components

• while keeping the overall complexity relatively the same - just remember that that mess of

• transistors and wires is still there.

• For example, another useful boolean operation in computation is called an Exclusive OR - or XOR for short.

• XOR is like a regular OR, but with one difference: if both inputs are true, the XOR is false.

• The only time an XOR is true is when one input is true and the other input is false.

• It’s like when you go out to dinner and your meal comes with a side salad OR a soup

• sadly, you can’t have both!

• And building this from transistors is pretty confusing, but we can show how an XOR is created

• from our three basic boolean gates.

• We know we have two inputs again -- A and B -- and one output.

• Let’s start with an OR gate, since the logic table looks almost identical to an OR.

• There’s only one problem - when A and B are true, the logic is different from OR,

• and we need to outputfalse”.

• If we add an AND gate, and the input is true and true, the output will be true.

• This isn’t what we want.

• But if we add a NOT immediately after this will flip it to false.

• Okay, now if we add a final AND gate and send it that value along with the output of our

• original OR gate, the AND will take infalseandtrue”, and since AND needs both values

• to be true, its output is false.

• That’s the first row of our logic table.

• If we work through the remaining input combinations, we can see this boolean logic

• circuit does implement an Exclusive OR.

• And XOR turns out to be a very useful component, and well get to it in another episode,

• so useful in fact engineers gave it its own symbol too -- an OR gate with a smile :)

• But most importantly, we can now put XOR into our metaphorical toolbox and not have to worry

• about the individual logic gates that make it up, or the transistors that make up those gates,

• or how electrons are flowing through a semiconductor.

• Moving up another layer of abstraction.

• When computer engineers are designing processors, they rarely work at the transistor level,

• and instead work with much larger blocks, like logic gates, and even larger components

• made up of logic gates, which well discuss in future episodes.

• And even if you are a professional computer programmer, it’s not often that you think

• about how the logic that you are programming is actually implemented in the physical world

• by these teeny tiny components.

• Weve also moved from thinking about raw electrical signals to our first representation

• of data - true and false - and weve even gotten a little taste of computation.

• With just the logic gates in this episode, we could build a machine that evaluates complex logic statements,

• like ifName is John Green AND after 5pm OR is Weekend

• AND near Pizza Hut”, thenJohn will want pizzaequals true.

• And with that, I'm starving, I'll see you next week.

Hi, I’m Carrie Anne and welcome to Crash Course Computer Science!

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# Boolean Logic & Logic Gates: Crash Course Computer Science #3

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黃齡萱 posted on 2017/07/05
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