Basic Logical Functions and Gates
While each logical element or condition must always have a logic
value of either "0" or "1", we also need to have ways to combine
different logical signals or conditions to provide a logical result.
For example, consider the logical statement: "If I move the switch
on the wall up, the light will turn on." At first glance, this
seems to be a correct statement. However, if we look at a few
other factors, we realize that there's more to it than this. In
this example, a more complete statement would be: "If I move the
switch on the wall up and the light bulb is good and
the power is on, the light will turn on."
If we look at these two statements as logical expressions and use
logical terminology, we can reduce the first statement to:
Light = Switch
This means nothing more than that the light will follow the action
of the switch, so that when the switch is up/on/true/1 the light
will also be on/true/1. Conversely, if the switch is
down/off/false/0 the light will also be off/false/0.
Looking at the second version of the statement, we have a slightly
more complex expression:
Light = Switch and Bulb and Power
Normally, we use symbols rather than words to designate the and
function that we're using to combine the separate variables of
Switch, Bulb, and Power in this expression. The symbol normally
used is a dot, which is the same symbol used for multiplication in
some mathematical expressions. Using this symbol, our
three-variable expression becomes:
|Light = Switch
When we deal with logical circuits (as in computers), we not only
need to deal with logical functions; we also need some special
symbols to denote these functions in a logical diagram. There are
three fundamental logical operations, from which all other
functions, no matter how complex, can be derived. These functions
are named and, or, and not. Each of these
has a specific symbol and a clearly-defined behavior, as follows:
- The AND Gate
- The AND gate implements the AND function. With the
to the left, both inputs must have logic 1 signals
applied to them in
order for the output to be a logic 1. With either input
at logic 0, the
output will be held to logic 0.
required for the demonstrations built into this page,
you will see buttons to the left of the AND gate
drawing. You can click on these to change their
assigned logic values, and the drawing will change to
reflect the new input states.
There is no limit to the number of inputs that may be
applied to an AND function, so there is no functional
limit to the number of inputs an AND gate may have.
However, for practical reasons, commercial AND gates
are most commonly manufactured with 2, 3, or 4 inputs.
A standard Integrated Circuit (IC) package contains 14
or 16 pins, for practical size and handling. A
standard 14-pin package can contain four 2-input
gates, three 3-input gates, or two 4-input gates, and
still have room for two pins for power supply
- The OR Gate
- The OR gate is sort of the reverse of the AND gate.
function, like its verbal counterpart, allows the output
to be true (logic
1) if any one or more of its inputs are true. Verbally,
we might say, "If
it is raining OR if I turn on the sprinkler, the lawn
will be wet." Note
that the lawn will still be wet if the sprinkler is on
and it is also
raining. This is correctly reflected by the basic OR
In symbols, the OR function is designated with a plus
sign (+). In logical
diagrams, the symbol to the left designates the OR
As with the AND function, the OR function can have any
number of inputs. However, practical commercial OR
gates are mostly limited to 2, 3, and 4 inputs, as
with AND gates.
- The NOT Gate, or Inverter
- The inverter is a little different from AND and OR
gates in that it always has exactly one input as well as
one output. Whatever logical state is applied to the
input, the opposite state will appear at the output.
The NOT function, as it is called, is necesasary in
many applications and highly useful in others. A
practical verbal application might be:
The door is NOT locked = You may enter
The NOT function is denoted by a horizontal bar over the
value to be inverted, as shown in the figure to the
left. In some cases a single quote mark (') may also be
used for this purpose: 0' = 1 and
1' = 0. For greater clarity in some logical
expressions, we will use the overbar most of the time.
In the inverter symbol, the triangle actually denotes
only an amplifier, which in digital terms means that
it "cleans up" the signal but does not change its
logical sense. It is the circle at the output which
denotes the logical inversion. The circle could have
been placed at the input instead, and the logical
meaning would still be the same.