Microsoft Word - Digital Logic Design v_4_6a

• For each pattern, if the independent variable value is 0, it is un-complemented, and if 1, it is
  complemented in the max-term which is the OR of all independent variables.
  For example: X=1, Y=1, Z=0  M 6 = X+Y+Z


• Each max-term will result in the output for that term being zero.

Here is an example for a 3-input system:

Step 1) Understand the problem

Write the expression for a function that is true when more than 1 input is true, otherwise the

function is 0.

Step 2) Develop a truth table for the function and write max-terms:

All independent variables must be present in each Max-term

* It is complemented if the variable value is 1.

* It is un-complemented if the variable value is 0.

Input

X Y Z

Standard Sum

Terms (Max-terms)

Max-term

Designators

Output

F

0 0 0 X+Y+Z M 0 F(0,0,0) = F 0 = 0

0 0 1 X+Y+Z M 1 F(0,0,1) = F 1 = 0

0 1 0 X+Y+Z M 2 F(0,1,0) = F 2 = 0

0 1 1 X+Y+Z M 3 F(0,1,1) = F 3 = 1

1 0 0 X+Y+Z M 4 F(1,0,0) = F 4 = 0

1 0 1 X+Y+Z M 5 F(1,0,1) = F 5 = 1

1 1 0 X+Y+Z M 6 F(1,1,0) = F 6 = 1

1 1 1 X+Y+Z M 7 F(1,1,1) = F 7 = 1

Step 3) Write the algebraic function equivalent to the truth table by rule:

For Compact Max-term Form:

If the Output function (F) is 0 for the max-term, then the value appears in the algebraic form

of the expression.

F(X,Y,Z) = (F 0 + M 0 ). (F 1 + M 1 ). (F 2 + M 2 ). (F 3 + M 3 ). (F 4 + M 4 ). (F 5 + M 5 ). (F 6 + M 6 ). (F 7 + M 7 )

= ( )

7

i 0 i i

Π F +M

=

Generalized compact max-term form of the function

Note the when F i=1, the max-term is not needed --- for our example:

F(X,Y,Z) = (0 + M 0 ). (0 + M 1 ). (0 + M 2 ). (1 + M 3 ). (0 + M 4 ). (1 + M 5 ). (1 + M 6 ). (1 + M 7 )

F(X,Y,Z) = M 0. M 1. M 2. M 4 Compact Max-term form of the function

Other forms:

F(X,Y,Z) = ΠM )4,2,1,0( Explicit Compact max-term form for 1s of the function

F(X,Y,Z) = Π )4,2,1,0( Implicit Compact max-term form for 1s of the function

The Not (Complement) of F can be written by writing the missing max-terms for Un-
complemented F: