Basic Logic Gates

e	s
$0$	$1$
$1$	$0$

In this lecture, we use the $\cdot$ operator to denote conjunction (the AND operation). It is not to be confused with multiplication of integers or reals. But, as will become more clear in the next chapter, both share some common properties. Another common symbol for the conjunction is $\land$ .

Symbol

The OR Gate

Description

The output is $0$ if and only if both inputs equal $0$ .²

Truth table

a	b	s
$0$	$0$	$0$
$0$	$1$	$1$
$1$	$0$	$1$
$1$	$1$	$1$

Equation

$s = a + b$

In this lecture, we use the $+$ operator to denote disjunction (the OR operation). It is not to be confused with addition of integers or reals (again, both share some common properties). Another common symbol for the conjunction is $\lor$ .

Symbol

The NAND (not AND) Gate

Truth table

Description

This is the complementary function of AND. The output is $0$ if and only if both inputs equal $1$ .

Truth table

a	b	s
$0$	$0$	$1$
$0$	$1$	$1$
$1$	$0$	$1$
$1$	$1$	$0$

Equation $s = \overline{a \cdot b}$

Symbol

The NOR (not OR) Gate

Description

This is the complementary function of the OR. The output is $1$ if and only if both inputs are $0$ .

Truth table

a	b	s
$0$	$0$	$1$
$0$	$1$	$0$
$1$	$0$	$0$
$1$	$1$	$0$

Equation $s = \overline{a + b}$

Symbol

Exclusive OR (XOR)

Description

The output is $1$ if one, and only one, of the two inputs is $1$ . ³

Truth table

a	b	s
$0$	$0$	$0$
$0$	$1$	$1$
$1$	$0$	$1$
$1$	$1$	$0$

Equation

$s = a \oplus b = a \cdot \overline{b} + \overline{a} \cdot b$

Symbol

Exclusive NOT OR (XNOR)

Description

This is the complementary function of the exclusive OR. The output is $1$ if and only if both inputs are equal.

Truth table

a	b	s
$0$	$0$	$1$
$0$	$1$	$0$
$1$	$0$	$0$
$1$	$1$	$1$

Equation

$s = \overline{a \oplus b} = a \cdot b + \overline{a} \cdot \overline{b}$

Symbol

The AND function can also be interpreted as a function of forcing a signal to zero: in the expression $s = v a l i d \cdot b$ , the signal $s$ is equal to $b$ only if $v a l i d = 1$ , otherwise it is 0.

The OR function can also be interpreted as a function of forcing a signal to one: in the expression $s = f orce + b$ , the signal $s$ is equal to $b$ only if $f orce = 0$ otherwise it is 1.

The exclusive OR function can also be interpreted as a selection function between a datum and its complementary: in the expression $s = se l co m p \oplus b$ , the signal $s$ is equal to $b$ if $se l co m p = 0$ otherwise it is $\overline{b}$

INF107

Basic logic functions

The Inverter (NOT)

The AND Gate

The OR Gate

The NAND (not AND) Gate

The NOR (not OR) Gate

Exclusive OR (XOR)

Exclusive NOT OR (XNOR)