Boolean simplification

Simplification of Boolean Expressions

Here are some examples of Boolean algebra simplifications. Each line gives a form of the expression, and the rule or rules used to derive it from the previous one. Generally, there are several ways to reach the result. Here are some of the examples for you as guideline :

• Simplify: AB(A + B)(B + B):

Expression	Rule(s) Used
AB(A + B)(B + B)	Original Expression
AB(A + B)	Complement law, Identity law.
(A + B)(A + B)	DeMorgan's Law
A + BB	Distributive law. This step uses the fact that or distributes over and. It can look a bit strange since addition does not distribute over multiplication.
A	Complement, Identity.

• Simplify: (A + C)(AD + AD) + AC + C:

Expression	Rule(s) Used
(A + C)(AD + AD) + AC + C	Original Expression
(A + C)A(D + D) + AC + C	Distributive.
(A + C)A + AC + C	Complement, Identity.
A((A + C) + C) + C	Commutative, Distributive.
A(A + C) + C	Associative, Idempotent.
AA + AC + C	Distributive.
A + (A + T)C	Idempotent, Identity, Distributive.
A + C	Identity, twice.

• Simplify: A(A + B) + (B + AA)(A + B):

Expression	Rule(s) Used
A(A + B) + (B + AA)(A + B)	Original Expression
AA + AB + (B + A)A + (B + A)B	Idempotent (AA to A), then Distributive, used twice.
AB + (B + A)A + (B + A)B	Complement, then Identity. (Strictly speaking, we also used the Commutative Law for each of these applications.)
AB + BA + AA + BB + AB	Distributive, two places.
AB + BA + A + AB	Idempotent (for the A's), then Complement and Identity to remove BB.
AB + AB + AT + AB	Commutative, Identity; setting up for the next step.
AB + A(B + T + B)	Distributive.
AB + A	Identity, twice (depending how you count it).
A + AB	Commutative.
(A + A)(A + B)	Distributive.
A + B	Complement, Identity.