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NAND, NOR and XOR Gates

Combined gates: NOT-AND, NOT-OR and exclusive OR — plus why NAND and NOR are called universal gates.

All Six Symbols

A NAND gate is an AND gate followed by a NOT. Its output is the opposite of AND : it is 0 only when all inputs are 1 . Expression: Q = A \cdot B . The bar goes over the whole of A \cdot B .

The NAND gate

Q = 1

NOT-AND: output is 0 only when every input is 1.

$Boolean expression: Q = A \cdot B$

Try it:

Compare each row with the AND gate — every output is flipped.

A NOR gate is an OR gate followed by a NOT. Its output is 1 only when all inputs are 0 . Expression: Q = A + B .

The NOR gate

Q = 1

NOT-OR: output is 1 only when every input is 0.

$Boolean expression: Q = A + B$

Try it:

Find the one row that gives 1.

An XOR gate gives 1 when the inputs are different, and 0 when they are the same. It is like OR, except it excludes the case where both are 1. Expression: Q = A \oplus B, which is the same as Q = A \cdot B + A \cdot B . An XNOR gate is XOR followed by NOT: 1 when the inputs are the same . Its expression is A \oplus B .

Logic gate explorer

Q = 0

Exclusive OR: output is 1 when the inputs are different .

$Boolean expression: Q = A \oplus B$

Try it:

Try XOR, then switch to XNOR. You can also revisit any gate from this course.

Truth tables for all two-input gates
Truth tables for all two-input gates

NAND is AND flipped; NOR is OR flipped; XNOR is XOR flipped.

OR vs XOR

$OR$	$XOR$
$1 if at least one input is 1$	$1 if the inputs are different$
$1, 1 \to 1$	$1, 1 \to 0$
$"Tea or coffee?" — you may have both$	$"Tea or coffee?" — pick exactly one$

NAND and NOR are called universal gates : you can build every other gate from only NAND gates (or only NOR gates). For example, a NAND gate with both inputs joined together acts as a NOT gate. Chip makers like this because a whole circuit can be made from one type of gate.

Practice

Select every gate whose output is 0 when A = 1 and B = 1.

Choose all that apply.

NAND — NOT AND

A NAND gate is an AND gate followed by a NOT. Its output is the opposite of AND : it is 0 only when all inputs are 1 . Expression: Q = A \cdot B . The bar goes over the whole of A \cdot B .

The NAND gate

Q = 1

NOT-AND: output is 0 only when every input is 1.

$Boolean expression: Q = A \cdot B$

Try it:

Compare each row with the AND gate — every output is flipped.

XOR — Exclusive OR

An XOR gate gives 1 when the inputs are different, and 0 when they are the same. It is like OR, except it excludes the case where both are 1. Expression: Q = A \oplus B, which is the same as Q = A \cdot B + A \cdot B . An XNOR gate is XOR followed by NOT: 1 when the inputs are the same . Its expression is A \oplus B .

Logic gate explorer

Q = 0

Exclusive OR: output is 1 when the inputs are different .

$Boolean expression: Q = A \oplus B$

Try it:

Try XOR, then switch to XNOR. You can also revisit any gate from this course.

Six Gates in One Table

Truth tables for all two-input gates
Truth tables for all two-input gates

NAND is AND flipped; NOR is OR flipped; XNOR is XOR flipped.

OR vs XOR

$OR$	$XOR$
$1 if at least one input is 1$	$1 if the inputs are different$
$1, 1 \to 1$	$1, 1 \to 0$
$"Tea or coffee?" — you may have both$	$"Tea or coffee?" — pick exactly one$

Universal Gates

NAND and NOR are called universal gates : you can build every other gate from only NAND gates (or only NOR gates). For example, a NAND gate with both inputs joined together acts as a NOT gate. Chip makers like this because a whole circuit can be made from one type of gate.

Practice

Select every gate whose output is 0 when A = 1 and B = 1.

Choose all that apply.