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De Morgan; NOR gate from NAND gates

Short answer

De Morgan's laws ease complementation, convert SOP to NAND-NAND and POS to NOR-NOR, and show gate equivalence. A NOR gate is built from four NANDs: two as inverters for A' and B', one to produce (A'B')' = A+B, and one inverter to yield (A+B)'.

What is the importance of De-morgan's laws? Show how a two-input NOR gate can be constructed from a two-input NAND gate.

Importance of De Morgan's laws

De Morgan's laws (A+B‾=AˉBˉ\overline{A+B} = \bar A\bar B, AB‾=Aˉ+Bˉ\overline{AB} = \bar A + \bar B) are used to:

  • complement Boolean expressions easily;
  • convert AND-OR (SOP) circuits to NAND-NAND and OR-AND (POS) circuits to NOR-NOR;
  • build every gate from one universal gate (NAND or NOR), reducing chip types and cost;
  • show gate equivalences such as NAND = bubbled OR and NOR = bubbled AND.

NOR gate from 2-input NAND gates

Required: Y=A+B‾Y = \overline{A + B}. By De Morgan, A+B=Aˉ⋅Bˉ‾A + B = \overline{\bar A \cdot \bar B}, so:

G1=A⋅A‾=AˉG2=B⋅B‾=BˉG3=Aˉ⋅Bˉ‾=A+BY=G3⋅G3‾=A+B‾\begin{aligned} G_1 &= \overline{A \cdot A} = \bar A \\ G_2 &= \overline{B \cdot B} = \bar B \\ G_3 &= \overline{\bar A \cdot \bar B} = A + B \\ Y &= \overline{G_3 \cdot G_3} = \overline{A + B} \end{aligned}

Four NAND gates are needed: two as input inverters, one to make OR, and one as the output inverter.

 A --[NAND G1]-- A' --+
   (inputs tied)      +--[NAND G3]-- A+B --[NAND G4]-- Y
 B --[NAND G2]-- B' --+                  (inputs tied)
ABAˉ\bar ABˉ\bar BA+BA+BY
001101
011010
100110
110010

✓ Answer checked by Claude Opus 5.5 (high effort)

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