Logic Gates
Every logic gate has one or more inputs and exactly one output, and can be described four equivalent ways: logic symbol, truth table, Boolean expression, timing diagram (see timing-diagrams for the last of these, and for real-gate propagation delay / rise / fall times).
The 7 basic gate types
| Gate | Behaviour | Truth table (\(A,B \to X\)) |
|---|---|---|
| NOT (inverter) | Inverts the input | 0→1, 1→0 |
| AND | \(X=1\) iff all inputs are 1 | 00→0, 01→0, 10→0, 11→1 |
| OR | \(X=1\) iff at least one input is 1 | 00→0, 01→1, 10→1, 11→1 |
| NAND | \(X=1\) iff at least one input is 0 (NOT AND) | 00→1, 01→1, 10→1, 11→0 |
| NOR | \(X=1\) iff all inputs are 0 (NOT OR) | 00→1, 01→0, 10→0, 11→0 |
| XOR | \(X=1\) iff exactly one input is 1 (“odd function”) | 00→0, 01→1, 10→1, 11→0 |
| XNOR | \(X=1\) iff inputs are the same (“even function”) | 00→1, 01→0, 10→0, 11→1 |
Useful to remember: XOR is the odd function, XNOR is the even function — this generalises directly to more than 2 inputs (XOR = 1 iff an odd number of inputs are 1).
Universal (complete) gates
All circuits can be constructed from NAND-only or NOR-only gates — these are called complete (or universal) gates, because NOT, AND, and OR can each be built purely from one gate type:
- NOT: a NAND (or NOR) gate with both inputs tied together.
- AND: two NANDs in series (NAND followed by another NAND used as an inverter), or the NOR equivalent.
- OR: built similarly from NAND-only or NOR-only combinations.
NAND and NOR are preferred in practice because they’re easier to build from transistors than AND/OR directly.
Gates on ICs
Example: the 74HCT00 integrated circuit packages four independent 2-input NAND gates into a single 14-pin DIP chip (Vcc = power e.g. 5V, GND = ground/0V; pin spacing 0.1”×0.3”, chip ~15mm long).