Algebraic Manipulation of Boolean Expressions - The Logic Tutorial

Algebraic Manipulation of Boolean Expressions

Boolean algebra is a fundamental concept in discrete structures and the theory of logic. It deals with mathematical operations applied to binary variables, represented by truth values - true (1) and false (0). Algebraic manipulation of Boolean expressions involves simplifying and transforming logical statements using algebraic rules and theorems.

Basic Operations

Boolean algebra operates on Boolean variables using three fundamental operations: AND, OR, and NOT.

• AND (·): The result is true only if both operands are true.
• OR (+): The result is true if at least one operand is true.
• NOT (¬): Negates the input; true becomes false and false becomes true.

Algebraic Laws

Several algebraic laws govern the manipulation of Boolean expressions:

• Identity Laws: A + 0 = A, A · 1 = A
• Domination Laws: A + 1 = 1, A · 0 = 0
• Idempotent Laws: A + A = A, A · A = A
• Commutative Laws: A + B = B + A, A · B = B · A
• Associative Laws: (A + B) + C = A + (B + C), (A · B) · C = A · (B · C)
• Distributive Laws: A · (B + C) = (A · B) + (A · C), A + (B · C) = (A + B) · (A + C)
• De Morgan's Laws: ¬(A + B) = ¬A · ¬B, ¬(A · B) = ¬A + ¬B

Algebraic Manipulation Techniques

To simplify Boolean expressions, several techniques are employed:

1. Substitution: Replace parts of the expression with equivalent terms based on algebraic laws.
2. Factoring: Group common terms to simplify the expression.
3. Consensus Theorem: (A · B) + (A̅ · C) + (B · C) = (A · B) + (A̅ · C)
4. Adjacency: A · (A + B) = A, A + (A · B) = A
5. Consensus: A + (B · C) + (A̅ · C) = A + B · C
6. Shannon's Expansion: F(A, B, C) = A · F(1, B, C) + A̅ · F(0, B, C)

Examples

Let's illustrate these concepts with examples:

1. Simplify the expression: A + A̅·B

Applying the idempotent law, A + A̅·B = A + B

2. Simplify the expression: (A + B)·(A̅ + C)

Using the distributive law, (A + B)·(A̅ + C) = (A · A̅) + (A · C) + (B · A̅) + (B · C)

Further simplification leads to A̅B + AC + AB̅ + BC

Using the consensus theorem, A̅B + AC + AB̅ + BC = A̅B + AC + BC