First thing I need to teach you is a new form of math, this is the math that digital circuits use and is surprisingly simple. The biggest difference between Arithmetic and Boolean, is that Arithmetic has an infinite number of states(3, 7.28, 18832, 0, -48123.28, etc) while Boolean only has two (0 and 1). 0 means simply nothing, it can also be understood as false, no, or in circuitry low signal. 1 on the other hand is anything, also understood as true, yes, or high signal.
I'm going to take you all the way back to 1st grade in order to help you understand this.
Arithmetic word problem: Jimmy had 3 apples, then Sally gives Jimmy 2 apples. How many apples does Jimmy have? Remember this one from when you were a little kid? Well its pretty clear to find the answer you would do 3+2=5. To make this a Boolean equation we need to make applesauce. Jimmy had applesauce, then Sally gives Jimmy applesauce. Does Jimmy have applesauce? To solve this we would have to see that Jimmy has applesauce, which means a 1, and Sally gives him more applesauce, again a 1. So the equation would be 1+1=1, Yes it equals 1 and not 2 because in Boolean we do not count, anything that is not 0 is a 1.
I hope now you have some basic understanding of what makes Boolean Algebra unique. Now because Boolean does not deal with quantity we only have 3 basic functions, Addition, Multiplication, and Inversion.
This works almost exactly the same as Arithmetic, except again we do not deal with quantity. To give an example:
0+1=1 Read this as "Nothing plus Something equals Something"
1+1=1 "Something plus Something equals Something"
0+0=0 "Nothing plus Nothing equals Nothing"
Note: 1 plus anything is 1
This is perhaps the smoothest transition because it works exactly like what we are used to seeing.
0x1=0 "No rows of some columns equals nothing"
1x1=1 "Some rows of Some columns equals something"
Note: 0 times anything is 0
Something we don't have an Algebraic equivalent to. Inversion simply takes a state and makes it the opposite. Inversion is normally marked with a line above the character, but for this I'm going to use ~ before.
~1=0 "Not Something is Nothing"
~0=1 "Not Nothing is Something"
This concludes the lesson on Boolean Algebra, and now onto how it relates to Digital Logic, and ultimately Redstone Circuits.
Digital Logic is the basis of what allows computers to run, it is composed of gates which I will be discussing below. This are not actual quotes, its just for organization.
The most basic function, it preforms the task of Inversion, which you just read about above. It can go by either name.
As seen on a schematic
Note the circle at the tip of the triangle, this signifies that it is an inverter, if you see this circle on any inputs or outputs of a logic device there is an inverter there.
This gate does multiplication, all inputs must be high for this to give a high output.
The gate for addition, if any input is high the output will also be high.
With these three basic gates you can build any digital device. For more info on this, and other common gates please visit the minecraftwiki page here.
In this final section I cover various components you can use to control your redstone circuitry.
0. Powered block-
While not an actually a redstone device it is important. A block powered by a button, lever, or pressure pad, will power any redstone that is next to, ontop of, or under the powered block. Blocks can also be powered by a straight piece of redstone wire, however it will not power additional redstone, this has to do with torches. More on this later
1. The button-
Buttons can only be placed on walls, when activated they turn on for about 1 second. They will power any block they are on, as well as any adjacent redstone, so any redstone 1 block under, or level with it and
2. Pressure Pad-
Pressure pads can only be placed on the ground and are activated by weight. They come in two varieties, stone which requires a player or NPC to stand on to activate, and wood which any object can hold down including boats, minecarts, or dropped items. Pressure pads power the block under them and any adjacent redstone.
Levers can be placed with 3 different orientations.
a) On a wall- Up is off, down is on. They function similar to a button except they can be left on.
b)On the ground (West-East)- West is off, East is on. These are similar to Pressure Pads in what they power, the block under them and all adjacent redstone.
c)On the ground (North-South)- North is on, South is off. These are unique as they do not power the block they are on, instead they only power adjacent redstone. This may be a bug.
The alignment of ground levers seems to be random, as I could not get any consistency.
4. Redstone Torches-
This is where digital logic becomes possible.
They can be placed on the ground, or on a wall. Torches power redstone adjacent to, or 1 block below them. They will also power the block directly above. Torches on a powered block, including by redstone wire, will turn off. This function is what makes them to act like an inverter and allows us to build more complex circuits.
I hope you guys learned something from this thread. Questions, comments, and suggestions are welcomed.
Pictures of Redstone logic gates: Minecraftwiki.net
Schematic Symbols of logic gates: Wikipedia.org