Boolean Algebra Laws to Know for Mathematical Logic

Boolean Algebra Laws are fundamental in Mathematical Logic, providing rules for manipulating logical expressions. These laws, like Commutative and Distributive, help simplify and clarify complex statements, making it easier to understand relationships between variables and their operations.

1. Commutative Laws

   • The order of variables does not affect the outcome of the operation.
   • For AND: A ∧ B = B ∧ A
   • For OR: A ∨ B = B ∨ A
   • This law simplifies expressions and allows for flexibility in rearranging terms.

2. Associative Laws

   • The grouping of variables does not change the result of the operation.
   • For AND: (A ∧ B) ∧ C = A ∧ (B ∧ C)
   • For OR: (A ∨ B) ∨ C = A ∨ (B ∨ C)
   • This law helps in simplifying complex expressions by changing the grouping of terms.

3. Distributive Laws

   • Describes how AND distributes over OR and vice versa.
   • A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)
   • A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)
   • This law is crucial for expanding and simplifying logical expressions.

4. Identity Laws

   • Establishes the identity elements for AND and OR operations.
   • A ∧ 1 = A (AND with true)
   • A ∨ 0 = A (OR with false)
   • These laws confirm that certain values do not change the outcome of operations.

5. Complement Laws

   • Each variable has a complement that, when combined, yields the identity element.
   • A ∧ ¬A = 0 (AND with its complement)
   • A ∨ ¬A = 1 (OR with its complement)
   • This law highlights the relationship between a variable and its negation.

6. Idempotent Laws

   • Repeated application of the same variable does not change the outcome.
   • A ∧ A = A
   • A ∨ A = A
   • This law simplifies expressions by eliminating redundant terms.

7. Absorption Laws

   • Shows how one operation can absorb another under certain conditions.
   • A ∧ (A ∨ B) = A
   • A ∨ (A ∧ B) = A
   • This law is useful for reducing expressions to their simplest form.

8. De Morgan's Laws

   • Provides a way to express negations of conjunctions and disjunctions.
   • ¬(A ∧ B) = ¬A ∨ ¬B
   • ¬(A ∨ B) = ¬A ∧ ¬B
   • These laws are essential for transforming logical expressions and understanding negation.

9. Double Negation Law

   • States that negating a negation returns the original value.
   • ¬(¬A) = A
   • This law reinforces the concept of negation in logical expressions.

10. Duality Principle

    • Every algebraic expression remains valid when AND and OR are interchanged, along with 0 and 1.
    • If an expression is true, its dual is also true.
    • This principle highlights the symmetry in Boolean algebra and aids in understanding logical equivalences.