Digital Logic Toolkit

Digital Logic Toolkit — Gates & Circuits

Source
https://digitallogictoolkit.com/logic-gates/
Printed

Logic gates and circuits

Eight gates, two symbol standards and the circuits built from them. Every truth value on this page is computed by the same engine that drives the tools it links to, so the reference and the calculator cannot disagree.

What is a logic gate?

A logic gate is a circuit element that takes one or more binary inputs — each of them either 0 or 1 — and produces exactly one binary output, determined only by the present combination of its inputs and not by anything that happened before. Each gate implements a single Boolean operation, so a truth table listing every input combination is its complete specification. There are seven in ordinary use plus the buffer, and every one of them can be built from NAND alone, or NOR alone.

Every two-input gate, side by side. Toggle A and B to trace a row.
ABANDORNANDNORXORXNOR
00001101
01011010
10011010
11110001
The one-input gates. B is ignored.
ANOTBuffer
010
101

The same eight gates written as algebra.

AND
A·B
OR
A + B
NOT
Ā
NAND
(A·B)′ = Ā + B̄
NOR
(A + B)′ = Ā·B̄
XOR
A ⊕ B = A·B̄ + Ā·B
XNOR
A ⊙ B = A·B + Ā·B̄
Buffer
A

The eight gates

Each card carries the gate’s symbol, its algebraic form and its complete truth table. The page behind it adds both symbol standards side by side, a live input toggle, the NAND-only and NOR-only builds and the worked examples.

  • AND gate

    &

    A·B

    AND truth table
    ABAND
    000
    010
    100
    111

    Output 1 only when every input is 1.

  • OR gate

    ≥1

    A + B

    OR truth table
    ABOR
    000
    011
    101
    111

    Output 1 when at least one input is 1.

  • NOT gate

    1

    Ā

    NOT truth table
    ANOT
    01
    10

    One input, inverted. Also called an inverter.

  • NAND gate

    &

    (A·B)′ = Ā + B̄

    NAND truth table
    ABNAND
    001
    011
    101
    110

    AND followed by an inverter, and universal on its own.

  • NOR gate

    ≥1

    (A + B)′ = Ā·B̄

    NOR truth table
    ABNOR
    001
    010
    100
    110

    OR followed by an inverter, and the other universal gate.

  • XOR gate

    =1

    A ⊕ B = A·B̄ + Ā·B

    XOR truth table
    ABXOR
    000
    011
    101
    110

    Output 1 when the inputs differ — the difference detector.

  • XNOR gate

    =

    A ⊙ B = A·B + Ā·B̄

    XNOR truth table
    ABXNOR
    001
    010
    100
    111

    Output 1 when the inputs match — the equality detector.

  • Buffer

    1

    A

    Buffer truth table
    ABuffer
    00
    11

    Output follows the input; it restores drive and adds delay.

Symbols: distinctive and rectangular

The distinctive shapes of ANSI/IEEE Std 91-1984 dominate teaching in the US and UK; the rectangular symbols of IEC 60617-12 dominate continental Europe and much of Asia. Handing in the wrong set costs marks, so pick yours here and every diagram on the site will use it.

Distinctive shapes — ANSI/IEEE Std 91-1984

  • AND
  • OR
  • NOT
  • NAND
  • NOR
  • XOR
  • XNOR
  • Buffer

Rectangular shapes — IEC 60617-12

  • &AND
  • ≥1OR
  • 1NOT
  • &NAND
  • ≥1NOR
  • =1XOR
  • =XNOR
  • 1Buffer

IEC 60617-12 is a paid standard whose per-symbol dimensions are not public. These rectangular symbols follow the published module-grid construction, so they are a reasonable construction rather than quoted metrics.

Symbols and universality

  • Logic gate symbols

    Every gate in both symbol standards, side by side, on a printable one-page chart.

  • Universal gates

    Rebuild any of the other gates using only NAND, or only NOR, with the gate count.

Building circuits

Related

The other two sections, and where they meet this one.

  • Boolean & Minimisation

    Turn one expression into every other form of itself, and reduce it with named laws.

  • Numbers & Codes

    Convert between bases and representations, and check the codes that detect and fix errors.