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Why NAND and NOR are universal gates

Short answer

NAND and NOR are universal because each can implement NOT, AND, and OR. Examples: NAND (A A) = A', (A NAND B) NAND (A NAND B) = AB, (A NAND A) NAND (B NAND B) = A+B. Thus any Boolean function can be built with only one type.

Why NAND and NOR gates are called Universal gates? Illustrate with examples.

NAND and NOR gates are called universal gates because each of them alone can realise the three basic operations NOT, AND and OR, and therefore any Boolean function. A complete circuit can thus be made using only one type of gate (e.g. 7400 quad NAND or 7402 quad NOR), which reduces inventory, cost and design effort.

Using NAND only

GateRealisationGates
NOTA⋅A‾=Aˉ\overline{A \cdot A} = \bar A1
ANDAB‾⋅AB‾‾=AB\overline{\overline{AB} \cdot \overline{AB}} = AB2
ORAˉ⋅Bˉ‾=A+B\overline{\bar A \cdot \bar B} = A + B3
 NOT: A --[NAND]-- A'      (inputs tied)
 AND: A,B --[NAND]--[NAND]-- AB
 OR : A --[NAND]-- A' --+
      B --[NAND]-- B' --+--[NAND]-- A+B

Using NOR only

GateRealisationGates
NOTA+A‾=Aˉ\overline{A + A} = \bar A1
ORA+B‾+A+B‾‾=A+B\overline{\overline{A+B} + \overline{A+B}} = A + B2
ANDAˉ+Bˉ‾=AB\overline{\bar A + \bar B} = AB3
 NOT: A --[NOR]-- A'
 OR : A,B --[NOR]--[NOR]-- A+B
 AND: A --[NOR]-- A' --+
      B --[NOR]-- B' --+--[NOR]-- AB

Example: Y=AB+CDY = AB + CD can be done with three NAND gates: Y=AB‾⋅CD‾‾Y = \overline{\overline{AB} \cdot \overline{CD}} (NAND-NAND form of an SOP).

✓ Answer checked by Claude Opus 5.5 (high effort)

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