Is there a proof for the FOIL method in Boolean algebra?

Question

The FOIL method is the special case of multiplying algebraic expressions using the distributive law and is shown here:

What does the proof for this look like using Boolean algebra?

Answer 1

As Boolean expression, this works out similarly:

Conjunctive Normal Form (CNF, product-of-sums):

(a+b) & (c+d)

Equivalent Disjunctive Normal Form (DNF, sum of products):

ac + ad + bc + bd

Karnaugh map:

             ab
       00  01  11  10
      +---+---+---+---+
   00 | 0 | 0 | 0 | 0 |
      +---+---+---+---+
   01 | 0 | 1 | 1 | 1 |
cd    +---+---+---+---+
   11 | 0 | 1 | 1 | 1 |
      +---+---+---+---+
   10 | 0 | 1 | 1 | 1 |
      +---+---+---+---+