Shannon's expansion theorem

In Boolean algebra, Shannon’s expansion theorem essentially states that we can decompose a logic function into the products of co-factors with respect to a given input.

$$f(x_0,x_1,\cdots ,x_i,\cdots ,x_n)={\bar{x}}_i\cdot f(x_0,x_1,\cdots ,0,\cdots ,x_n)+x_i\cdot f(x_0,x_1,\cdots ,1,\cdots ,x_n)$$

Decomposition

We choose a variable to isolate for and find its cofactors.[^1]

$$f=x\bar{y}+xyz+\bar{x}y\bar{z}+\bar{y}z$$

Then:

$$f_x=\bar{y}+yz+\bar{y}z$$ $$f_{\bar{x}}=y\bar{z}+\bar{y}z$$

which corresponds to: If we find the $z$ or $y$ cofactors of the $x$ cofactors, then we end up with 4 different equations, in terms of only one variable. What this means is that we can put it in a 2-to-1 mux as followed:

$$f_{xz}=\bar{y}+y+\bar{y}=1$$ $$f_{x\bar{z}}=\bar{y}$$ $$f_{\bar{x}z}=\bar{y}$$ $$f_{\bar{x}\bar{z}}=y$$

That’s about all to know about this.