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We have learned how to specify digital systems,

now let's see how we can implement it.

The system implementation, means the procedure of how to build a bul,

a digital system from smaller components.

And these smaller components was simple components with building blocks, can

be basic logical gates, can be universal gates which we are going to define later,

or can be some indus, industrial technology library, we can,

we can use, always use flip-flop to, to store memory.

And also, nowadays when people use, reuse space of design,

they're talking about using intellectual properties as the building block.

[SOUND] Now let's see how we can implement Boolean functions using basic

logical gates, by basic logical gates, we are talking about the three gates,

the AND gate, which represents the Boolean AND, x and the y,

which will be true if and only if both x and y are true, the logical OR,

x plus y, which will be true even if any one of x and y will be true, and,

the inverter, NOT gate, x prime, which will change a true to a false.

And with this three basic logical gate,

we can implement all kind of Boolean expressions, for example,

this w based on three or four variables, x0, x1, x2, x3,

it is defined, as the sum of four terms, each term is a logical end, and

from here, we can easily draw the logical implementation of this function.

So for example here, we have x0, and x1, so this AND

gate implements this term, and then the last AND

gate takes x3, and x0 so this one implements the third term.

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And this OR gate, implements the sum, of this, four terms.

[SOUND] And besides the three basic logical gates,

there are several other logi, logical gates.

So for example, the NAND gate, the NOR gate, Exclusive-OR gate, and

Exclusive-NOR gate.

The functionality of these gates are given in this truth table, and

on these first two truth columns, we list all the four combinations of two inputs,

x and the y, start from 0, 0, 0, 1, go to 1, 0, then 1, 1, and then for each

of these input combinations, we list, the output, for each of these four functions.

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It has three input x, y, and z and then two output, S and

a C, and this basically implements a which is going to add up x and

y plus a carry z, and S is the new sum, and the C is the new carry.

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So from this definition, we can also draw the logical diagram of this function.

So input x, y, z, output sum and a carry, and the inside, we have the basic logical

gates, gate, gate, OR gate and the two Exclusive-OR gate.

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So how we do this is, we start, from the definition of Boolean functions,

we've realized that, all of the Boolean functions can be expressed as the logical

AND, the logical OR, or

inverters, so by putting these three gates together, we have the so

called, the standard universal gate, so all the Boolean functions,

should be able to implemented by these three logical gates.

And now this seems becomes easier, so for us to prove, any logical gate

is universal, we only need to show that this cage can be used to implement

the standard universal gates which is logical AND, logical OR, and the inverter.

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Let's see in the example here.

[SOUND] We define a logical gate as a Boolean function f of x,

y, z, it is defined as, as x prime yz plus xy prime plus y prime z,

and we claim this function with this gate is universal.

And how do we prove this?

The first one will show that, I can use this function block f or

this gate to implement the inverter.

And how do I do that?

I set, the second and the third parameter to f as constant to 1,

so if I plug in this into the definition of f, I got x prime

times 1 times 1 plus x times 1 compliments plus 1 compliments times 1.

And because 1 compliments equal to 0, so this last of two terms disappeared,

and then 1 times anything is, that thing itself,

so this give us x compliments, that is exactly the inverter.

And to draw a, a diagram, I can show this, so this is my logical gate f, and

how I feed into this system is I feed x and 1, 1 into f.

And, but higher from outside the box, what people see is,

I give it input x to the system, and then the system outputs x prime,

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So next, we show you how to implement the OR gate, and

then similarly, we put the middle, parameter to 0, and

then plug in to the definition of f, so we, what we get is x prime and a y 0,

so this becomes 0 times z, and the z is the new y here.

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And plus the first variable, which is x, times the compliments of

the second variable, which is zero compliments, plus the compliments of

the second variable times the third variable, which is y here.

And because we know that 0 complements is 1, and 0, cancel this one so

the only thing left is x plus y, which is the logical OR.

'Kay?

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And then to show this concept, we can go the same figure, is thus the box you

feed the x and the y, and internally you are taking 0 as the middle, parameter,

let this function f, give you x plus y, so from outside, this looks like a OR gate.

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And here is the logical imp, implementation here,

with two functional blocks, this one, the first one, get y complements,

the second one get x times y, and from the outside you see x, y coming in,

and then from outside you see x times y coming out, so, this is the logical AND.