강의노트 논리 회로의 연산과 정리

부울 대수의 주요 정리 및 법칙

항등 법칙

$$\begin{align*} \mathrm{A} + 0 &= \mathrm{A} \\ \mathrm{A} \cdot 1 &= \mathrm{A} \end{align*}$$

소멸자

$$\begin{align*} \mathrm{A} + 1 &= 1 \\ \mathrm{A} \cdot 0 &= 0 \end{align*}$$

멱등 법칙

$$\begin{align*} \mathrm{A} + \mathrm{A} &= \mathrm{A} \\ \mathrm{A} \cdot \mathrm{A} &= \mathrm{A} \end{align*}$$

보수 (역원)

$$\begin{align*} \mathrm{A} + \overline{\mathrm{A}} &= 1 \\ \mathrm{A} \cdot \overline{\mathrm{A}} &= 0 \end{align*}$$

이중 부정

$$\overline{\overline{\mathrm{A}}} = \mathrm{A}$$

교환 법칙

$$\begin{align*} \mathrm{A} + \mathrm{B} &= \mathrm{B} + \mathrm{A} \\ \mathrm{A} \cdot \mathrm{B} &= \mathrm{B} \cdot \mathrm{A} \end{align*}$$

결합 법칙

$$\begin{align*} \mathrm{A} + \left( \mathrm{B} + \mathrm{C} \right) &= \left( \mathrm{A} + \mathrm{B} \right) + \mathrm{C} \\ \mathrm{A} \cdot \left( \mathrm{B} \cdot \mathrm{C} \right) &= \left( \mathrm{A} \cdot \mathrm{B} \right) \cdot \mathrm{C} \end{align*}$$

흡수 법칙

$$\begin{align*} \mathrm{A} \cdot \left( \mathrm{A} + \mathrm{B} \right) &= \mathrm{A} \\ \mathrm{A} + \left( \mathrm{A} \cdot \mathrm{B} \right) &= \mathrm{A} \end{align*}$$

분배 법칙

$$\begin{align*} \mathrm{A} \cdot \left( \mathrm{B} + \mathrm{C} \right) &= \mathrm{A} \cdot \mathrm{B} + \mathrm{A} \cdot \mathrm{C} \\ \mathrm{A} + \left( \mathrm{B} \cdot \mathrm{C} \right) &= \left( \mathrm{A} + \mathrm{B} \right) \cdot \left( \mathrm{A} + \mathrm{C} \right) \end{align*}$$

드모르간의 정리

$$\begin{align*} \overline{ \mathrm{A} + \mathrm{B}} &= \overline{\mathrm{A}} \cdot \overline{\mathrm{B}} \\ \overline{ \mathrm{A} \cdot \mathrm{B}} &= \overline{\mathrm{A}} + \overline{\mathrm{B}} \end{align*}$$