2-8
The first combination (A = 0, B = 0) corresponds to T0 in figure 2-4; the second to T1; the third to T2;
and the last to T4. When constructing a Truth Table, you must include all possible combinations of the
inputs, including the all 0s combination.
A Truth Table representing an AND gate with three inputs (X, Y, and Z) is shown below. Remember
that the two-input AND gate has four possible combinations, with only one of those combinations
providing a HIGH output. An AND gate with three inputs has eight possible combinations, again with
only one combination providing a HIGH output. Make sure you include all possible combinations. To
check if you have all combinations, raise 2 to the power equal to the number of input variables. This will
give you the total number of possible combinations. For example:
EXAMPLE 1-AB = 22 = 4 combinations
EXAMPLE 2-XYZ = 23 = 8 combinations
X
Y
Z
f
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
0
1
0
0
0
1
0
1
0
1
1
0
0
1
1
1
1
f = XYZ
As with all AND gates, all the inputs must be HIGH at the same time to produce a HIGH output.
Don’t be confused if the complement of a variable is used as an input. When a complement is indicated as
an input to an AND gate, it must also be HIGH to satisfy the input requirements of the gate. The Boolean
expression for the output is formulated based on the TRUE inputs that give a TRUE output. Here is an
adage that might help you better understand the AND gate:
In order to produce a 1 output, all the inputs must be 1. If any or all of the inputs is/are 0, then the
output will be 0.
Referring to the following examples should help you cement this concept in your mind. Remember,
the inputs, whether the original variable or the complement must be high in order for the output to be
high. The three examples given are all AND gates with two inputs. Keep in mind the Boolean expression
for the output is the result of all the inputs being HIGH.
