The Boolean algebra is used to describe certain logic design made up with several logic gates and wires connecting them.

Laws and Theorems in Boolean Algebra

Basic Theorems

• $A+B=B+A$, and $AB=BA$.

• $(A+B)+C=A+(B+C)$, and $(AB)C=A(BC)$.

• $A(B+C)=AB+AC$.

• $A+AB=A$, and $A(A+B)=A$.

  This theorem shows that $B$ will be ‘absorbed’ in this form of expression.

• $AB+A\bar{B}=A$, and $(A+B)(A+\bar{B})=A$.

• $A+\bar{A}B=A+B$.

  $A+\bar{A}B=A+AB+\bar{A}B=A+(A+\bar{A})B=A+B$.

• ${A+BC=(A+B)(A+C)}$

  Why: $(A+B)(A+C)=A+AC+BA+BC$, and $A+AC+BA=A$.

• $AB+\bar{A}C+BC=AB+\bar{A}C$, and $AB+\bar{A}C+BCD=AB+\bar{A}C$.

  Use $BC$ to absorb $BCD$.

• $\overline{A\bar{B}+\bar{A}B}=\bar{A}\bar{B}+AB$.

  XNOR = XOR + NOT.

DeMorgan’s Law

$$ \begin{align*}\overline{X_1X_2\cdots X_n}=\bar{X}_1+\bar{X}_2+\cdots+\bar{X}_n\\\overline{X_1+X_2+\cdots+X_n}=\bar{X}_1\bar{X}_2\cdots\bar{X}_n\end{align*} $$

Dual Rule

By changing operators in an expression in this way you get its dual expression:

• $+\leftrightarrow\cdot$
• $\oplus\leftrightarrow\odot$

For a equation it’s dual expression is still valid. For example: