Q.Explain the universal property of NAND gate and realize the AND, OR, NOT and X-OR gates with their respective truth tables.

The NAND gate is called a universal gate because every basic logic function — NOT, AND, OR (and hence any circuit, including X-OR) — can be built using NAND gates alone.

[!ANSWER]

NOT = $\overline{A\cdot A}=\overline{A}$ (1 NAND); AND = $\overline{\overline{AB}}=AB$ (2 NANDs); OR = $\overline{\overline{A}\,\overline{B}}=A+B$ (3 NANDs); X-OR = 4 NANDs giving $A\overline{B}+\overline{A}B$ — so NAND is universal.

[!TLDR]

NAND is universal: NOT, AND, OR and X-OR can all be realised with NAND gates only.

Universal property: A gate is called universal if the three basic gates (NOT, AND, OR) can be constructed from it alone — and hence any logic function. The NAND gate satisfies this, so any digital circuit can be built entirely from NAND gates. This is economical because only one type of IC need be stocked.

1) NOT gate — tie both NAND inputs together: $Y=\overline{A\cdot A}=\overline{A}$.

2) AND gate — a NAND followed by a NAND-inverter: $Y=\overline{\overline{AB}}=A\cdot B$ (2 NANDs).

3) OR gate — invert both inputs with NAND-inverters, then NAND them: $Y=\overline{\overline{A}\cdot\overline{B}}=A+B$ (3 NANDs, by De Morgan).

4) X-OR gate — four NANDs: $G1=\overline{AB}$; $G2=\overline{A\cdot G1}$; $G3=\overline{B\cdot G1}$; $Y=\overline{G2\cdot G3}=A\overline{B}+\overline{A}B=A\oplus B$.

[!ANSWER]

NAND is universal because NOT $=\overline{A\cdot A}=\overline{A}$, AND $=\overline{\overline{AB}}=AB$, OR $=\overline{\overline{A}\,\overline{B}}=A+B$ and X-OR $=\overline{G2\cdot G3}=A\overline{B}+\overline{A}B$ are all realisable with NAND gates alone, each verified by the truth tables above.