Laws of Boolean Algebra: Simplifying Logical Expressions
Master the fundamental laws governing Boolean algebra—the algebra of logic. This guide details key theorems, including commutative, associative, distributive, and De Morgan's laws, providing the tools to simplify and manipulate Boolean expressions for efficient digital logic design.
Laws of Boolean Algebra
Boolean algebra is a type of algebra that deals with binary variables (variables that can only have two values, typically 0 and 1, representing false and true) and logical operations (AND, OR, NOT).
Fundamental Laws of Boolean Algebra
Several fundamental laws govern Boolean algebra. These laws help simplify and manipulate Boolean expressions, making them easier to work with.
1. Commutative Law
The order of operands doesn't change the result:
- A + B = B + A (OR)
- A * B = B * A (AND)
2. Associative Law
The grouping of operands doesn't change the result:
- A + (B + C) = (A + B) + C (OR)
- A * (B * C) = (A * B) * C (AND)
3. Distributive Law
Multiplication distributes over addition (and vice versa):
- A + (B * C) = (A + B) * (A + C)
- A * (B + C) = (A * B) + (A * C)
4. Annulment Law
- A * 0 = 0
- A + 1 = 1
5. Identity Law
- A * 1 = A
- A + 0 = A
6. Idempotent Law
- A + A = A
- A * A = A
7. Complement Law
- A + A' = 1
- A * A' = 0
(Where A' represents the complement of A—the opposite value.)
8. Double Negation Law
(A')' = A
9. Absorption Law
- A + (A * B) = A
- A * (A + B) = A
De Morgan's Laws
De Morgan's Laws describe how to negate compound expressions. They are based on the principle of duality (swapping AND and OR and 0 and 1).
- ¬(A ∨ B) = ¬A ∧ ¬B (The complement of an OR is the AND of the complements)
- ¬(A ∧ B) = ¬A ∨ ¬B (The complement of an AND is the OR of the complements)
Conclusion
These laws are fundamental to Boolean algebra and are used extensively in simplifying and manipulating logical expressions, which are critical in digital electronics and computer science.