Calculators look at a list of inputs and produce
an output. Computers do the same thing. We use truth
tables to show this list of inputs and outputs.
Inputs are limited to two possibilities, either 0 or
1. This is the meaning of digital. The name digital
(or binary) means two values. A 0 is called False
and a 1 is called True.

Theory

The simplest truth table is for one input and one
output. In this case there are two possible truth
tables. These are shown below.

Input

Output

0

0

1

1

Figure A

Input

Output

0

1

1

0

Figure B

For Figure A it is the same as having a wire
connecting the input and the output. Whatever
appears on the input is transferred to the output.

For Figure B, the opposite happens.
This is called a NOT operation (NOT gate). The
output is not the input. The output is True (1) if
the input is not True (1). The output is False (0)
if the input is not False (0)

Another way to look at a truth table is as a list
of possible events and what will happen in each
case. Suppose your friend John might come to visit.
If he comes you will not watch Matlock. If he does
not come you will watch Matlock. Then the table
would look like the following.

Event

Result

John Comes

You Don't Watch Matlock

John Doesn't Come

You Watch Matlock

NOT GATE

We will build a device with one input and one
output. We will give the device an input, either
True (1) or False (0). The device will give us an
output (either True or False) to tell us what
happened as a result of our input. Later we will
build devices with more inputs and outputs so it can
do more complex things for us.

When we build this first device we will use 0
volts for False (0) and we will use 5 volts for True
(1). If the input to our device is 0 volts then the
output will be 5 volts. If the input is 5 volts then
the output will be 0 volts. This will be our first
logic gate, the not gate. It is also called an
inverter. Logic gates are devices that are built to
do truth tables. Computer chips like the Pentium are
made of logic gates.

AND GATE

Now lets look at a device with two inputs and one
output. First, we will look at a device that does an
AND operation. Our inputs will be A and B. Our
output will be C.

If A and B are both True (1) then the output, C,
will be True (1). Otherwise C will be False (0).
This operation is shown in the truth table given
below.

Input A

Input B

Output

Input A

Input B

Output

False

False

False

0

0

0

False

True

False

0

1

0

True

False

False

1

0

0

True

True

True

1

1

1

The table gives a list of all possible
combinations of inputs and the resulting output for
each combination. This is not the only possible
combination of outputs, but this paticular
combination is called an AND gate.

OR GATE

Another possibility for a device with two inputs
and one output is an OR operation. For inputs A and
B and output C : If A or B is True (1) then C is
True (1). Otherwise C is False (0).

To build the truth table, we first look at the
inputs. A can be True or False. B can be True or
False. First we make a table of all possible
combinations of A and B.

A

B

False

False

False

True

True

False

True

True

Now we can determine the output by saying for
each combination, if A or B is true then the output,
C, is True. Then we have the following table.

A

B

C

False

False

False

False

True

True

True

False

True

True

True

True

Then to convert this to a form we can build
(using 5 Volts for True and 0 Volts for False) we
make the following table.

A

B

C

A

B

C

False (0V)

False (0V)

False (0V)

0V

0V

0V

False (0V)

True (5V)

True (5V)

0V

5V

5V

True (5V)

False (0V)

True (5V)

5V

0V

5V

True (5V)

True (5V)

True (5V)

5V

5V

5V

Boolean Algebra

These three gates (the NOT gate, the OR gate, and
the AND gate) are the basic building blocks of
digital design. They are all that is needed to build
the most complex computers that exist. To build
complex designs, a type of math has been developed
to deal with binary numbers. It is called Boolean
Algebra. It gives you a way to combine these three
gates into bigger designs.

The three basic operations have symbols. A NOT
operation is represented by a line over a letter.
Instead of using this line we will just say ‘bar’.
For example Abar means 'not A'. So if Abar is 1 then
A is 0. If Abar is 0 then A is 1.

An OR operation is represented by a + sign. For
example, A + B = A OR B.

An AND operation is represented by a *. For
example, A * B means A AND B.

Examples

1.) If A = 1 (5V) and B = 0 (0V) then what is A *
B?

Since 1 is a True and 0 is a False then we can
say the problem is ~ True * False = ?. If we put in
AND for * then the problem is ~ True AND False = ?.
To answer this we can look back at the truth table
for the AND operation and see that True AND False is
False. Another way to find the answer is to look at
the definition of the AND operation. It says if A
and B are True then the output is True. Otherwise
the output is false. In the above example since B is
False the output is False so A * B = False (A * B =
0).

2.) If A = 1 (5V) and B = 0 (0V) then what is A +
B?

To do this problem we can say that A = 1 = True
and B = 0 = False. Then the problem is ~ True OR
False = ? We can either look at the truth table for
the OR operation or we can look at the definition of
the OR operation which says. If A or B is True then
the output is True. Otherwise the output is False.
Since A is True in our question then the output is
True. So True + False = True (1 + 0 = 1).