Overview
This lecture covers three key theorems in Boolean algebra: the Uniting Theorem, the Elimination Theorem, and the Absorption Theorem, with a focus on their statements, dual forms, and proofs.
Uniting Theorem
- The theorem states: X AND (Y OR Y') = X.
- The dual form: X OR (Y AND Y') = X.
- Proof uses the fact that (Y OR Y') = 1 and (Y AND Y') = 0.
- Factoring X in expressions simplifies terms to X.
Elimination Theorem
- The theorem states: X OR (X' AND Y) = X OR Y.
- The dual form: X AND (X' OR Y) = X AND Y.
Absorption Theorem
- The theorem states: X OR (X AND Y) = X.
- The dual form: X AND (X OR Y) = X.
Key Terms & Definitions
- Dual — The form of a Boolean statement after interchanging AND (⋅) and OR (+) and swapping 0 and 1.
- Prime (') — Indicates logical NOT or complement (e.g., X' means NOT X).
- Absorption — Boolean rule where combining a term with its conjunction/disjunction yields the original term.
Action Items / Next Steps
- Review and practice applying these theorems to simplify Boolean expressions.
- Prepare to use these rules in upcoming logic circuit problems.