The Consensus Theorem is a boolean algebra simplification rule used to eliminate redundant terms without changing the output of a boolean expression.
- It simplifies logic expressions and reduces the number of logic gates.
- Helps design efficient digital circuits.
Consensus Theorem in SOP and POS Forms
The Consensus Theorem can be expressed in both Sum-of-Products (SOP) and Product-of-Sums (POS) forms:
- Sum-of-Products (SOP) Form
AB + A'C + BC = AB + A'C
- Product-of-Sums (POS) Form
(A + B)(A' + C)(B + C) = (A + B)(A' + C)
Here, A' represents the complement of A. In both SOP and POS forms, the consensus term is redundant and can be removed without affecting the output of the Boolean expression.
Proof of the Consensus Theorem
We prove the SOP form step by step:
- Start with the left-hand side:
AB + A'C + BC
- Add consensus term:
AB + A'C + BC = AB + A'C + BC(A + A') (since A + A' = 1)
- Expand BC(A + A'):
AB + A'C + ABC + A'BC
- Factor out common terms (Absorption Law):
AB + ABC + A'C + A'BC
AB(1 + C) + A'C(1 + B)
AB + A'C ---> (since 1 + C = 1 and 1 + B = 1)
- Final simplification:
AB + A'C + BC = AB + A'C
Thus, the term BC is not needed.
Solved Examples
Example 1: Simplify the following Boolean expression using the Consensus Theorem:
F = AB + BC' + AC
Here, the expression satisfies the conditions of the Consensus Theorem. Variable C appears in both complemented (C') and non-complemented (C) forms, and AB is the consensus (redundant) term.
According to the Consensus Theorem, the consensus term AB is redundant and can be removed without affecting the output. Therefore, the simplified expression is:
∴ F = BC' + AC
Example 2: Apply the Consensus Theorem to simplify:
F = (A + B) (A' + C) (B + C)
Here, the expression satisfies the conditions of the Consensus Theorem. Variable A appears in both complemented (A') and non-complemented (A) forms, and (B + C) is the consensus (redundant) term.
According to the Consensus Theorem, the consensus term (B + C) is redundant and can be removed without affecting the output. Therefore, the simplified expression is:
∴ F = (A + B) (A' + C)
Advantages
- Simplifies Boolean Expressions: Eliminates redundant terms without changing the output.
- Reduces Logic Gates: Fewer gates simplify circuit implementation.
- Lowers Hardware Cost: Reduced gate count decreases circuit complexity and cost.
Disadvantages
- Limited Applicability: Works only when the expression satisfies the theorem's conditions.
- Requires Pattern Recognition: Identifying the consensus term may not be straightforward.
- Cannot Simplify Every Expression: Many Boolean expressions require other simplification methods.
Applications
- Logic Circuit Simplification: Reduces the number of logic gates in digital circuits.
- Boolean Function Minimization: Simplifies Boolean expressions during digital circuit design.
- Digital System Design: Used in the design of CPUs, multiplexers, decoders, and other digital systems.
