Positive/negative logic; verify 3-variable De Morgan

Positive and negative logic

• Positive logic: HIGH voltage = logic 1, LOW voltage = logic 0 (e.g. TTL: +5 V = 1, 0 V = 0).
• Negative logic: HIGH voltage = logic 0, LOW voltage = logic 1 (e.g. +5 V = 0, 0 V = 1).

The same physical gate performs a different logic function in the two systems, because changing the convention complements every input and output. Example: a gate whose output is HIGH only when both inputs are HIGH is an AND gate in positive logic but an OR gate in negative logic.

De Morgan's theorem for 3 variables

Theorem 1: $\overline{A + B + C} = \bar A \cdot \bar B \cdot \bar C$

Theorem 2: $\overline{A \cdot B \cdot C} = \bar A + \bar B + \bar C$

Verification by truth table

Columns 4 and 5 are identical, and columns 6 and 7 are identical for all 8 combinations, so both theorems hold for 3 variables.

Algebraic check (using the 2-variable law with $X = B + C$):

$$\begin{aligned} \overline{A + B + C} &= \overline{A + X} = \bar A \cdot \bar X \\ &= \bar A \cdot \overline{B + C} = \bar A \bar B \bar C \end{aligned}$$

Similarly, $\overline{A(BC)} = \bar A + \overline{BC} = \bar A + \bar B + \bar C$.

 3-input NOR = bubbled 3-input AND
 3-input NAND = bubbled 3-input OR