Boolean Expression Simplification

• Simplify: C + BC:

  Expression	Rule(s) Used
  C + BC	Original Expression
  C + (B + C)	DeMorgan's Law.
  (C + C) + B	Commutative, Associative Laws.
  T + B	Complement Law.
  T	Identity Law.

• 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.

  You can also use distribution of or over and starting from A(A+C)+C to reach the same result by another route.
• 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