In my digital electronics labs and lectures we are told to try and make things from NAND gates, because they are the cheapest kind of gate available to buy. Why is this? Why isn't an OR/AND gate the cheapest to buy?
Digital Logic – Why Are NAND Gates Cheap?
digital-logic
Related Solutions
If your book is any good, it should mention Karnaugh mapping. Karnaugh mapping is about grouping like values (like all ones or all zeros) on a map consisting of rows and columns of logical combinations.
If you can group logical combinations like that you can simplify your logical function as given in this example.
Another method which works better for me is to work with truth tables. On the left you write all logical combinations of the inputs. For 3 inputs that would be
A B C
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Note that this table becomes rather long if you have many inputs: \$2^N\$ lines. On the right you write the output(s). For example
A B C Y
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 0
1 1 1 1
With some exercise you often can see a pattern in the output. In this case for the first four lines, where A = 0, Y = B XOR C. For the rest, where A = 1, Y = NOT (B XOR C), or, combined: Y = A XOR B XOR C. (This can be used to create a parity bit)
Jeff mentions how you can use DeMorgan's Law to create an OR gate from NANDs. This XOR gate is also basic:
I've turned this into a community wiki so we can collect cool logic gate implementations to which to refer to in the future.
To start with you have to understand transistors in a simple way. I will deal with CMOS since 99% of all logic that has ever existed (in number count) exists as CMOS.
There are two kinds of transistors used, PMOS and NMOS, here are their symbols:
The transistors are electrically controlled current sources/sinks. The PMOS will source current (the dotted line in the diagram shows current flow when on) from a power supply (attached to source) through the drain and into other circuits when the Gate voltage is LOWER than the source. The NMOS will sink current into ground through the drain into the source (which in this case you should think of as a sink).
Please note that I've taken some liberties with naming for the sake of clarity.
PMOS is usually connected to a positive voltage and NMOS is usually connected to negative voltages typically ground.
Interestingly you can stack the devices to make various functions. Stacking two PMOS gives a current source that is controlled by two voltages, stacking two NMOS gives a current sink that is controlled by two voltages.
Notice that both Voltage at A (we'll call it A) and B BOTH have to be below +V for current to flow. Also notice that Both C and D have to be higher than Ground (that funny hatched triangle symbol) for current to be sinked (sunk ?). You could say "Both A AND B have to low for current to flow" and "Both C AND D have to be high for current to flow".
Just like you can "stack" (actually put in series), you can parallel devices.
You could say that "either A OR B can be low for current to flow" for the PMOS and you could say that "either C OR D can be high for current to flow" for the NMOS circuit.
You will notice that already we are using logic language to describe function (AND, OR) so now we can start piecing together circuits.
First off the Invertor:
When Vin is at ground, the PMOS is turned on and can source current, but the NMOS is off and cannot sink current. As a result, the Vout pin tries to put charge onto any available capacitance and charges that capacitance up until it reaches the V+ level.
Likewise when the Vin is High, the NMOS is turned on and can sink current, but the PMOS is now off and cannot source current. as a result, the Vout pin tries to pull charge off of any available capacitance and discharges that capacitance until it reaches the Ground level.
A "high" on the input gives a "low" on the output, a "low" on the input gives a "high" on the output. It inverts!
If you look at the symbol for both the PMOS and the NMOS you'see that the gate looks like a capacitor on the symbol. This is deliberate as a MOS transistor IS a capacitor and it is mainly this capacitance that is charged and discharged during operation. Current is the flow of charge per time and capacitance is the storage of charge per voltage. Transistors turn gate voltage into controlled currents that then charge and discharge gate capacitances that turns that change in charge back into a change in voltage.
Now for the first two input gate the NAND gate:
The NMOS "stack" will only ever sink current under one condition, and that is when BOTH A & B are high. Notice that for that condition that BOTH the PMOS are off (i.e. do not source current). So in that condition Vout will sink current and the Vout will be Low.
In all other conditions at least one of the PMOS will be sourcing current and the NMOS stack will not be able to sink current. The output is then charged up and Vout = high.
A B Out
0 0 1
0 1 1
1 0 1
1 1 0
This truth table shows that if Not(A&B) AKA NAND. 0 = gnd, 1 = V+.
To turn into a AND gate you just need to invert the output.
And it's truth table:
A B Out
0 0 0
0 1 0
1 0 0
1 1 1
And next the NOR gate
I hope by now you should be able to get the truth table yourself.
C D Out
0 0 1
0 1 0
1 0 0
1 1 0
There is pleasing symmetry from NOR to NAND. the structure is a simple inversion.
Now the OR
and truth table
C D Out
0 0 0
0 1 1
1 0 1
1 1 1
Extending the designs to higher order inputs is easy, as shown by a 3 input NAND.
By placing NMOS and PMOS in series/parallel combinations you can implement various logic functions at a transistor level. This is often done for area efficiency, power efficiency or even for speed. These functions need not be strictly AND, OR or Xor functions. The following is known as as a AND/OR gate:
and has the following truth table.
C A B Out
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
Least you think that this is the only way to implement these functions, I'll introduce a device that is called the transmission gate.
Both the NMOS and PMOS gates must be driven in opposition to work properly.
Here is a sample circuit of what you can do with an additional NMOS.
Here /A = Not(A) in digital logic
A+B = A OR B
A*B = A AND B
So you can see that only using 3 transistors you can implement A OR B. BE warned though that this circuit has serious side effects and is not generally used. But it is illustrative nevertheless.
here is a whole collection of TG based logic functions:
There is also Pass-Transistor-Logic or PTL. An example of such: 
Best Answer
NAND gates are cheap because there are so many of them lying around from the 1980s.
Seriously though, a NAND gate is about the simplest logic gate. You can think of it as a multi-input inverter. Electrically, that's exactly what TTL NAND gates are. Each input is just another emitter added to the input transistor. The rest of the circuit is just a inverter. It's different in CMOS, but a NAND gate is still very simple.
Since the chips require few transistors, they can be small, which allows lots of them per silicon wafer, which makes them cheap.