Hi,
I have attached a picture to make it easier to ask my question. There is a table in the picture. I want to know what that table is called and how to use it. It looks like a karnaugh map to me but how do they get the x, x', 0s, and 1s in it.

Thank you for helping me in advance,
Hamid A.

 

They are both regular Karnaugh maps, which describe the sum and the carry out of an addition between the 1-bit numbers x and y, and the carry in named z. The MUX is set up to simulate the function of a Full Adder.

S is the sum and C is the carry out. X+Y+Z=CS
For example: 1+0+1=10, where we have X=1, Y=0, Z=1, C=1 and S=0.

The Karnaugh map contains all of the possible results for all the X,Y,Z combinations. Because the circuit is built with a 4-to-1 MUX, to help you a bit, the vertical labels of the map are named after the according input of the MUX which corresponds to a Y,Z combination.
\(I_0\) is the input of the MUX that corresponds to Y'Z'. \(I_1\) corresponds to Y'Z and so on.
This is especially helpful, because by simplifying the map only one column at a time, you can find what input you need to "give" the MUX in order to produce the correct result.
In this case, we can see that for the function of C, we need to input 0 at the \(I_0\) pin.

An alternative to find the MUX inputs without the Karnaugh map, can be found in post #5 in this thread:
http://forum.allaboutcircuits.com/showthread.php?t=53799&highlight=multiplexer+boolean+function

Is that clear?