Overview
This lecture continued the introduction to Boolean Algebra, focusing on key rules, including distributive, commutative, associative, and De Morgan's laws, along with an example problem simplification.
Review of Previous Rules
- Complement rule: a and a' = 0; a or a' = 1.
- AND rules: a and a = a; a and 0 = 0; a and 1 = a; a and a' = 0.
- OR rules: a or a = a; a or 0 = a; a or 1 = 1.
Distributive Law
- Distributive law: a and (b or c) = (a and b) or (a and c).
- Dual distributive law: a or (b and c) = (a or b) and (a or c).
- Important for simplifying Boolean expressions; frequently appears in exam problems.
- For a or a' b, use distributive law to simplify to a or b.
- Similarly, a' or a b simplifies to a' or b.
Commutative Law
- a or b = b or a.
- a and b = b and a.
Associative Law
- (a and b) and c = a and (b and c).
- (a or b) or c = a or (b or c).
Operation Priority
- NOT (complement) has highest priority, then AND, then OR.
De Morgan's Law
- (a or b)' = a' and b'.
- (a and b)' = a' or b'.
Example Problem Simplification
- Given expression: Y = a c' b or a c' b' or b c'.
- Factor common terms: a c' (b or b') or b c'.
- (b or b') = 1, so Y = a c' or b c'.
- The result uses fewer logic gates, demonstrating simplification.
Key Terms & Definitions
- Complement (NOT) — Flips value: a' is the opposite of a.
- AND Operation — Logical multiplication, only true if all inputs are true.
- OR Operation — Logical addition, true if at least one input is true.
- Distributive Law — Allows distribution of AND/OR over OR/AND.
- Commutative Law — Order does not affect outcome for AND/OR.
- Associative Law — Grouping does not affect outcome for AND/OR.
- De Morgan's Law — Rules for negating AND/OR combinations.
Action Items / Next Steps
- Write down and memorize all Boolean Algebra rules explained.
- Practice solving additional Boolean Algebra problems.
- Prepare for next presentation covering more example problems.