Introduction to the Algebra of Boole.

There are 256 possible logic circuits that can be constructed with 3 binary inputs and one binary output. Each circuit can be represented as a polynomial in the 3 inputs x, y, and z:
f(x,y,z) = a + bx + cy + dxy + ez + fxz + gyz +hxyz
There are 8 outputs of the circuit, corresponding to the 8 possible combination of inputs:f000, f001, f010, f011, f100, f101, f110, f111
The Universal Circuit calculates the 8 parameters {a,b,c,d,e,f,g,h} to the appropriate values to match the 8 outputs. This allows it to exactly emulate any one of these 256 possible circuits.