Microsoft Word - Digital Logic Design v_4_6a

 Let’s take another example from problem statement to truth table to min-terms and the resulting

sum of products.

Step 1) Understand the problem

Write out an expression for the function that is true, when 2 out of 3 inputs are true. Output is

false for all other input combinations.

Step 2) Develop a truth table for the function

Input

X Y Z

Standard Product

Terms (min-terms)

Min -term

Designators

Output

F

0 0 0 XY.. Z m 0 F(0,0,0) = F 0 = 0

0 0 1 XY.. Z m 1 F(0,0,1) = F 1 = 0

0 1 0 XY.. 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 XYZ m 7 F(1,1,1) = F 7 = 0

Note:

1) The min-term subscript corresponds to the binary value of the input.

2) All three independent input variables are present in each min-term.

3) When input is 1, the corresponding variable appears in the Min-term, otherwise the

variable is complemented in the min-term.

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

If the output function (F) is 1 for the “min-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

0

(. )

i

Fimi Generalized compact Min-term form of the function

= 0.m 0 + 0.m 1 + 0.m 2 + 1.m 3 + 0.m 4 + 1.m 5 + 1.m 6 + 0.m 7

F(X, Y, Z) = m 3 + m 5 + m 6 Compact min-term form of the function

F(X, Y, Z) = ∑m )6,5,3( Explicit Compact Min-term form for 1s of the function

F(X, Y, Z) = ∑ )6,5,3( Implicit Compact Min-term form for 1s of the function

By the way, the Not (Complement) of F can be written as (write the missing min-terms):

F (X, Y, Z) = ∑m )7,4,2,1,0( Explicit Compact Min-term form for 0s of the original function

F(X, Y, Z) = ∑ )7,4,2,1,0( Implicit Compact Min-term form for 0s of the original function

 Obtaining the Standard Products of Sum (POS) Form of Functions
Although POS is not used as much, there are times where the POS form is more efficient than SOP.

As the name applies, all three independent variables are present in either complemented or un-
complemented form.