How Boolean Logic Works

Implementing Gates

In the previous sections we saw that, by using very simple Boolean gates, we can implement adders, counters, latches and so on. That is a big achievement, because not so long ago human beings were the only ones who could do things like add two numbers together. With a little work, it is not hard to design Boolean circuits that implement subtraction, multiplication, division... You can see that we are not that far away from a pocket calculator. From there, it is not too far a jump to the full-blown CPUs used in computers.

So how might we implement these gates in real life? Mr. Boole came up with them on paper, and on paper they look great. To use them, however, we need to implement them in physical reality so that the gates can perform their logic actively. Once we make that leap, then we have started down the road toward creating real computation devices.

The easiest way to understand the physical implementation of Boolean logic is to use relays. This is, in fact, how the very first computers were implemented. No one implements computers with relays anymore -- today, people use sub-microscopic transistors etched onto silicon chips. These transistors are incredibly small and fast, and they consume very little power compared to a relay. However, relays are incredibly easy to understand, and they can implement Boolean logic very simply. Because of that simplicity, you will be able to see that mapping from "gates on paper" to "active gates implemented in physical reality" is possible and straightforward. Performing the same mapping with transistors is just as easy.

Let's start with an inverter. Implementing a NOT gate with a relay is easy: What we are going to do is use voltages to represent bit states. We will define a binary 1 to be 6 volts and a binary 0 to be zero volts (ground). Then we will use a 6-volt battery to power our circuits. Our NOT gate will therefore look like this:

[If this figure makes no sense to you, please read How Relays Work for an explanation.]

You can see in this circuit that if you apply zero volts to A, then you get 6 volts out on Q; and if you apply 6 volts to A, you get zero volts out on Q. It is very easy to implement an inverter with a relay!

It is similarly easy to implement an AND gate with two relays:

Here you can see that if you apply 6 volts to A and B, Q will have 6 volts. Otherwise, Q will have zero volts. That is exactly the behavior we want from an AND gate. An OR gate is even simpler -- just hook two wires for A and B together to create an OR. You can get fancier than that if you like and use two relays in parallel.

You can see from this discussion that you can create the three basic gates -- NOT, AND and OR -- from relays. You can then hook those physical gates together using the logic diagrams shown above to create a physical 8-bit ripple-carry adder. If you use simple switches to apply A and B inputs to the adder and hook all eight Q lines to light bulbs, you will be able to add any two numbers together and read the results on the lights ("light on" = 1, "light off" = 0).

Boolean logic in the form of simple gates is very straightforward. From simple gates you can create more complicated functions, like addition. Physically implementing the gates is possible and easy. From those three facts you have the heart of the digital revolution, and you understand, at the core, how computers work.

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