Lesson 0110: Proving Equalities

There are many uses for Boolean Algebra. One of them is testing equalities. This is done by taking an equality: and breaking it down into steps.
In this case, that expression is .

Proving equalities involves simplifying the more complicated expression until it is the same as the simpler expression, or until it is completely simplified.
However, like in a proof, reasons have to be given for every step taken when simplifying the more complicated expression.

AB + AB =	1. The reason for this is that it is "given".
AB + (A[not B] + [not A] B) =	2. The reason for this is the rule of [A xor B]
AB + A[not B] + [not A] B =	3. The reason for this is the commutative property
B(A+[not A]) + A[not B] =	4. The reason for this is the distribuitive property
B + A[not B] =	5. The reason for this is the property that states that A + [not A] = 1
A+B=	6. The reason for this is the property that states that B+[not B]A=A+B
7. Therefore AB + AB = A+B