peter taylor
- Joined Apr 1, 2013
- 106
Oh, and non-intuitively, all logic can be described using NAND (or NOR) gates.
You could put it that way, or say that AND and OR will allow (the input) current to pass through the load, in all but one input combination.This is not really what I have been saying.
The reason that these gates are fundamental is that from the four combinations of input 00, 01, 10 and 11, there is only ONE unique output, either 0, 0, 0, 1 for AND, or 0, 1, 1, 1 for OR.
XOR is not fundamental because it has two outcomes for four combinations of input, 0, 1, 1, 0 in this case.
He's not saying that it is -- just as you have gone off on a path that has nothing to do with the original question, so has he. This happens in threads when the TS doesn't keep the wanderings in check.This is not really what I have been saying.
You have yet to say what is so special about a gate being "fundamental", so at this point that is merely just a word that you have chosen to describe this condition. You could just as easily call just gates "lonely".I think that theoretically, all logic circuits can be made from NOT gates and AND gates, or NOT gates and OR gates, because each is a fundamental logic unit.
For instance, take an AND (or OR) gate with TWO inputs and ONE output, and a NOT gate with ONE input and ONE output (a two input NOT gate is nonsensical).
The reason that these gates are fundamental is that from the four combinations of input 00, 01, 10 and 11, there is only ONE unique output, either 0, 0, 0, 1 for AND, or 0, 1, 1, 1 for OR.
XOR is not fundamental because it has two outcomes for four combinations of input, 0, 1, 1, 0 in this case.
Says who? You've said before that if you don't like something that you are free to redefine it however you want.Now, a THREE input AND (or OR) gate also has ONE unique outcome from EIGHT combinations of input.
Point #1 is irrelevant.That is, 1 for an input 111, and a 0 for all other inputs 000, 001, 010, 011, 100, 101 and 110 (in binary notation, and inputs A, B and C are in any order).
The reason that a THREE input AND gate seems to be a logical extension of a TWO input AND gate is that:
1/ It still has one unique output, and:
2/ It still follows our basic concept of an AND function.
Now you are contradicting yourself. You keep making this big deal about how AND and OR gates are somehow "fundamental" because they truth table has one unique output, but now you have another gate that obeys that same definition and yet it is for some reason not "fundamental". So, once again, what in your world makes a gate "fundamental" and why is that important, interesting, or useful?I would't say it was FUNDAMENTAL though, because it can be made from two, TWO input AND gates.
What about the other four? I've asked you about this previously and you just responded with a sad face and the proceeded to ignore it.What makes AND and OR and NOT useful, is that each in combination can describe the four unique outcomes of a TWO input gate.
That is, for TWO inputs of 00, 01, 10, 11, in any order: OR gives 0, 1, 1, 1: AND gives 0, 0, 0, 1: NOR gives 1, 0, 0, 0: NAND gives 1, 0, 0, 0.
Complete non sequitur -- how does this claim follow in any way from the prattle that came before it?This makes it intuitive when designing simple logic. And there will always be a place in circuit design where a handful of gates will do a better job than a Micro Control Unit or a Large Scale Integration (LSI) Gate Array.
So? Now you are saying that "fundamental" means that we can use it to implement all other logic. We already have a word for that -- it is called a "universal gate". But you say that AND and OR are "fundamental", they that are NOT universal gates and you cannot implement all other logic with them alone. So, once again, just what do you mean when you claim a gate is "fundamental"? You are all over the map.Now, we can choose, If We Like, to reduce this combination of OR, AND, NOR and NAND down to a SINGLE FUNDAMENTAL UNIT, say, for instance, a NAND gate, because, ultimately, all other logic can be made from this, including an XOR gate.
You cannot make a NAND gate for XOR gates (and if someone can, I would be interested to know).
Consider yourself corrected because you are wrong.Practically though, and correct me if I'm wrong, but don't manufactures of LSI devices in some cases use ONLY NAND gates on there silicon real-estate. N-channel MOSFET's are faster than P-channel (because the current mechanism uses electron flow rather than hole flow), they are simpler to populate using current methods, resistors are made from them, they inherently follow NAND and NOT in there optimal circuit configuration, and because ONLY NAND's are used, complexity is minimized.
This was established long before you joined the thread. But nothing that you have added has really done anything to establish this conclusion independently.To sum up:
Yes, it is useful to have a universal definition of NOT, OR, AND and XOR, but
No, you cannot intuitively scale XOR to three or more inputs like AND and OR.
I suspected that this has been your primary goal for quite some time.I hope I have confused the thread
So? All logic can be describe using material implication, material nonimplication, converse implication, and converse nonimplication, as well. What of it?Oh, and non-intuitively, all logic can be described using NAND (or NOR) gates.
That's not at all a precise definition of a 2-input XOR gate. By this "precise" definition, ALL of the following are two input XOR gates:OK
An 2-i/p XOR gate is precisely defined as having one outcome half of the time, and a different outcome the other half of the time.
This is just babble.Intuitively, if one input AND the other input are different, then the output will be true, which KIND of makes it exclusive AND.
More babble.Then I would imagine that a three input XOR's output would be true if one, OR the other, OR the other input is different from the other two.
More babble.AND, I would imagine that a three input XOR's output would be true if one, AND the other, AND the other input is different from the other two.
Sounds like a good idea. Revisit things once you sober up.I might go and lie down now
Did you even bother reading, say, Post #2 in this thread?I think I am getting to the crux of the question, by defining what one would consider as BASIC and UNCHANGING and scale-able, so that a AND and an OR can have more than 2 inputs and maintain their basic, function, while an XOR can't.
Promise?I'm going to a different thread
While I agree in that I don't see the usefulness of categorizing gates by whether they have a unique output, I don't follow the rest of what you are saying. By "unique", Peter clearly meant that only one of the possible input combinations produces a particular output and all other input combinations produce the other. If that unique output is a 1, then there are four gates that have that behavior: NOR, Converse nonimplication, Material nonimplication, and AND. If that unique output is a 0, then there are four gates that have that behavior: NAND, Material implication, Converse Implication, and OR.I don't think it's useful to define gates by their unique output. In the case of AND, that output is 1. Being unique, there can't be another, and all of the input states have been exhausted. Then, by use of the 'other' unique output, 0, the same gate becomes OR.
Yes, only one of the four unique entries in the AND truth table, produces the output 1. How many uniques does one need?While I agree in that I don't see the usefulness of categorizing gates by whether they have a unique output, I don't follow the rest of what you are saying. By "unique", Peter clearly meant that only one of the possible input combinations produces a particular output and all other input combinations produce the other. If that unique output is a 1, then there are four gates that have that behavior: NOR, Converse nonimplication, Material nonimplication, and AND. If that unique output is a 0, then there are four gates that have that behavior: NAND, Material implication, Converse Implication, and OR.
And your point would be....?Then it is an Odd Parity Generator.
You have to define which operator is used when no operator is given. In normal math it is defined to be multiplication. In Boolean algebra, if it is allowed at all (some practitioners don't allow it and require the operators to appear explicitly) then it is the AND operation.Extensible: able to be extended.
Can you put brackets around terms x and y, thus (x)(y), and expect it to equal (xy).
In pure mathematics, yes.