## Wednesday, October 9, 2013

### 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