There are at least two paths to demonstrate a theorem: the classic algebraic method and perfect induction case, very useful in Boolean Algebra.

This last path says that if you check the veracity of a theorem for all possible input combinations, then the theorem is true in its entirety. This is, if it is fulfilled in each case, it is fulfilled in general. This path can be used in Boolean Algebra since the variables have only two possible values: 0 and 1, whilst in our algebra each variable can have infinite values.

For example, to demonstrate the distributed property of the sum against the product (which is not fulfilled in common algebra). X+(Y·Z) = (X+Y)·(X+Z)

 

X	Y	Z	X+(Y·Y)	(X+Y)·(X+Z)
0	0	0	0	0
0	0	1	0	0
0	1	0	0	0
0	1	1	1	1
1	0	0	1	1
1	0	1	1	1
1	1	0	1	1
1	1	1	1	1

In the table you can see that all possible values for X, Y and Z, the expressions X+(Y·Z) y (X+Y)·(X+Z) are identical, and thus, by perfect induction, both expressions are equivalent. The mechanism of demonstration by perfect induction is very useful in Boolean Algebra.