Logic gates

A logic gate is the base element of circuits and implements a boolean function.

Each gate takes one or more input and produces one output that is either HIGH (value 1) or LOW (value 0).

Base gates

name function description
AND \(f=x \cdot y\) HIGH when all inputs are HIGH, otherwise LOW
OR \(f=x+y\) HIGH when at least one input is HIGH
NOT (inverter) \(f = \overline x\) HIGH when the input is LOW, or LOW when the input is HIGH

Each of these base gates implement a core operation of Boolean algebra.

Other gates

name function description
XOR \(f=x \oplus y = x \cdot \overline y + \overline x \cdot y\) HIGH only when the inputs differ
NAND \(f = x \uparrow y = \overline{x \cdot y}\) equivalent to “not AND”
NOR \(f = x \downarrow y = \overline{x + y}\) equivalent to “not OR”
XNOR \(f = x \odot y = \overline{x \oplus y}\) equivalent to “not XOR”

Multiple inputs

AND, OR, XOR, and XNOR are all commutative and associative, they may be extended trivially to more than two inputs.

NAND and NOR operations are both commutative, but are not associative.

NOT always operates on a single input.