universal gates

universal gates

1. What Universal Gates Are

Universal gates = gates from which any Boolean function can be implemented using only that gate type, without needing other gates. The two universal gates are NAND and NOR.

• NAND = AND + NOT:

(bubble on AND)

• NOR = OR + NOT:

Both can realize NOT, AND, OR themselves, hence any function.

2. How NAND is Universal (show constructions)

NOT from NAND: Y = \overline{A·A} = \bar{A} (tie inputs together)

AND from NAND: Y = \overline{\overline{A·B}·\overline{A·B}} = A·B (NAND + NOT via NAND)

OR from NAND: Y = \overline{\bar{A}·\bar{B}} = A+B (De Morgan: \bar{A} via NAND, then NAND)

Similarly:

NOT from NOR: Y = \overline{A+A} = \bar{A}

OR from NOR: Y = \overline{\overline{A+B}} = A+B (NOR + NOT)

AND from NOR: Y = \overline{\bar{A}+\bar{B}} = A·B

3. Why Universal Matters

• Manufacturing: fabricate only one gate type (NAND in TTL, NOR in CMOS) → cheaper, denser.
• Logic design: any circuit can be built with NAND only (NAND logic).
• Truth tables:
  • NAND: $0·0→1, 0·1→1, 1·0→1, 1·1→0$ (opposite of AND)
  • NOR: $0+0→1, others 0$

4. Worked Illustration

Implement Y= A·B + C with NAND only:
1. Y= \overline{\overline{A·B}·\overline{C}}? Actually need OR: Y= \overline{\overline{A·B}·\bar{C}} using NANDs for NOT and AND.

Implement with NOR: similar.

Need to convert any expression to NAND-NAND or NOR-NOR form using De Morgan.

5. Other Gates (for context, not universal alone)

• XOR ($Y=A⊕B = A\bar{B}+\bar{A}B$), XNOR — not universal alone (need extra).

6. Exam Points

• Define universal gate and state NAND and NOR are universal — 3-5 marks, very frequent (almost certain to ask "which gates are universal?").
• Show NOT, AND, OR from NAND (and from NOR) with diagrams — 5-7 marks, very frequent (the most common universal gates question).
• Give truth tables for NAND/NOR — 3-5 marks, frequent.
• Mention De Morgan's use — 3 marks.