Boole's expansion theorem

Boole's expansion theorem is a way of combining two equations together to make one equation. To do this, it is helpful to think of equations like pieces of a puzzle; the equation is a picture of lots of little pieces, and you are trying to figure out how all of them fit together.

To use Boole's expansion theorem, you start with two equations that have the same number of pieces. Then, you line up the pieces so that each piece from one equation is next to a piece from the other equation. Lastly, you add up the pieces on each side, to create one equation with all the same pieces.

For example, if we have two equations that look like puzzles with four pieces

Puzzle A:

(A + B) + (C + D)

Puzzle B:

(P + Q) + (R + S)

We line up the pieces like this

(A + B) + (C + D)

(P + Q) + (R + S)

Then add up each side to get one new equation with all pieces:

(A + B + P + Q) + (C + D + R + S)

This is Boole's formula. You can use it to combine any two equations that have the same number of pieces.