Overview
This lecture introduces the Consensus Theorem in Boolean algebra, states both forms, and demonstrates their proofs using Boolean rules.
Consensus Theorem 1 (Sum Form)
- The first consensus theorem is: AB + A’C + BC = AB + A’C.
- AB + A’C + BC(A+A’) how and when?
- To prove, expand terms: AB + A’C + BCA + BCA’.
- Group and factor: AB(1 + C) + A’C(1 + B).
- Apply Boolean rules: 1 + C = 1, 1 + B = 1, so expression simplifies to AB + A’C.
- The result matches the right-hand side (RHS), confirming the theorem.
Consensus Theorem 2 (Product Form)
- The second consensus theorem is: (A + B)(A’ + C)(B + C) = (A + B)(A’ + C).
- Expand and simplify using Boolean law (e.g., AA’ = 0).
- Simplified terms: ABC + BA’ + BC + AC + A’BC.
- Combine like terms, especially BC, and factor if possible.
- After simplification, product is BC + BA’ + AC.
- RHS also simplifies to AC + BA’ + BC, matching LHS and proving the theorem.
Boolean Algebra Rules Used
- 1 + X = 1 (Dominance law).
- AA’ = 0 (Complement law).
- X + X = X and XX = X (Idempotent laws).
Key Terms & Definitions
- Boolean Algebra — mathematical system for logical operations with binary variables.
- Consensus Theorem — a principle stating redundant terms in Boolean expressions can often be eliminated.
- LHS/RHS — Left-Hand Side / Right-Hand Side of an equation or expression.
- Complement Law — a rule stating a variable ANDed with its complement is zero.
Action Items / Next Steps
- Review previous session on Boolean rules for deeper understanding.
- Prepare for upcoming sessions with competitive exam questions on Boolean equations.