Boolean algebra

Boolean algebra

1. What Boolean Algebra Is (for Digital)

Boolean algebra = algebra for binary variables ($0/1$ or false/true) with operations AND (·), OR (+), NOT (¯), obeying postulates parallel to ordinary algebra but with idempotence and complementation. Basis for minimizing logic circuits.

2. Axioms / Laws (memorize 8 with dual)

Law	Form (OR)	Form (AND)	Dual
Commutative	$A+B=B+A$	$A·B=B·A$	swap +↔·
Associative	$(A+B)+C=A+(B+C)$	$(A·B)·C=A·(B·C)$
Distributive	$A·(B+C)=A·B+A·C$	$A+B·C=(A+B)(A+C)$ (unlike ordinary!)
Identity	$A+0=A$, $A+1=1$	$A·1=A$, $A·0=0$
Idempotent	$A+A=A$	$A·A=A$
Complement	$A+\bar{A}=1$	$A·\bar{A}=0$
Involution	$\bar{\bar{A}}=A$
Absorption	$A+A·B=A$, $A·(A+B)=A$
De Morgan's	$\overline{A+B}= \bar{A}·\bar{B}$	$\overline{A·B}= \bar{A}+\bar{B}$	key for universal

3. Simplification Example

Y = A·B + A·\bar{B} = A(B+\bar{B})=A·1=A ⇒ needs one wire, not two ANDs + OR.

Y = (A+B)(A+C)=A+BC (distributive variant)

Use Karnaugh map for 3+ variables, but Boolean laws suffice for exam.

Prove: \overline{A·B+C}=? Use De Morgan stepwise.

4. Duality Principle

Dual of expression: swap +↔·, 0↔1, keep variables — if law holds, dual holds (e.g., $A+1=1$ dual is $A·0=0$).

5. Exam Points

• List laws (commutative, associative, distributive, De Morgan) — 5-7 marks, very frequent (almost certain to ask laws).
• State De Morgan's $\overline{A+B}=\bar{A}\bar{B}$, $\overline{AB}=\bar{A}+\bar{B}$ — 5 marks, very frequent.
• Simplify $A+AB=A$, $A(A+B)=A$, $AB+A\bar{B}=A$ — 5-7 marks, very frequent (common simplification numerical: minimize $Y=\bar{A}B+AB$).
• Show $A+\bar{A}B=A+B$ — 5 marks, frequent.