Logic to Switches
Logic via Electrical Switches? (Slide 9)
With the new tools from Boole and DeMorgan, others began to see where they could be applied in the real world. In 1886, Charles Sanders Peirce noted in a letter that logical operations could be easily simulated by electrical switches. Many others worked on the idea too many to name here, but 51 years later.
Claude Shannon (Slide 10)
In 1937, something happened. In 1937, Claude Shannon, a 21 year old graduate student at MIT was working on this same idea. He wrote a master’s thesis that some of called the most important master’s thesis of all time, titled a symbolic analysis of relay and switching circuits. In it, he showed that you could use electrical switches in Boolean algebra to construct circuits that could show any logical OR numerical relationship that you wanted. It’s available free online and will be linked in Canvas and linked in this video as well if you’re interested, but this is really cool. Kind of what was the gateway to electrical circuits, all of the cell phones and computers and electronic devices that you use today, this was the initial theory behind how all of those devices actually work.
Logic Gates (Slide 11)
So underneath Claude Shannon’s representation as far as translating Boolean algebra and Boolean logic into electrical circuits, we also needed a new representation for that this new representation is called logical gates. So it’s basically the same setup as we had with Boolean algebra and is very similar to if we have a and b, this is equivalent to this right where our two inputs are coming in on these two lines here are the AND operator it’s like a D, and then our output is the line leaving from the operator. similar idea for or so this is a or b we have XOR a XOR B, and not so this is same thing as saying that. Now really with the knot, the really important part is this little knot or this little mini circle at the end here and also note right that all of our Boolean operators here are compound operators, meaning that we have a left hand side and a right hand side, but the NOT operator only has one input. So one fact so just keep that in mind. But we can also apply the NOT operator to all of our other operators as well. So we can have NAND, nor, and x nor a little.at. The end here on the output is really the only part that matters for negating the result of a operation and and or XOR. One of the interesting things though, that it’s really kind of come out of the representation of Boolean logic on electrical circuits. So The work done by Claude Shannon and will continue part of this work and future lectures as well.
The Universal Logic Gate (Slide 12)
But one of the really interesting things that have kind of come out of this is the idea of a universal logic gate. The idea here is that any electrical circuit can be finished off by just using NAND gates. So all the complex Boolean logic that we could ever think of so any logical statements that we could write in Boolean algebra or Boolean logic can be redone with just using NAND gates or NAND operators, which is a pretty interesting thing and really why this is so important is that it greatly increases the speed efficiency and decreases the cost of manufacturing electrical parts.