1. ECE 301 – Digital Electronics Minterm and Maxterm Expansions and Incompletely Specified Functions (Lecture #6) The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6 th Edition, by Roth and Kinney, and were used with permission from Cengage Learning.

2. Spring 2011ECE 301 - Digital Electronics2 Minterms and Maxterms

3. Spring 2011ECE 301 - Digital Electronics3 Minterm In general, a minterm of n variables is a product (ANDing) of n literals in which each variable appears exactly once in either true or complemented form, but not both.  A literal is a variable or its complement. For a given row in the truth table, the corresponding minterm is formed by  Including the true form a variable if its value is 1.  Including the complemented form of a variable if its value is 0.

4. Spring 2011ECE 301 - Digital Electronics4 Minterms

5. Spring 2011ECE 301 - Digital Electronics5 Minterm Expansion When a function f is written as a sum (ORing) of minterms, it is referred to as a minterm expansion or a standard sum of products.  aka. “canonical sum of products”  aka. “disjunctive normal form” If f = 1 for row i of the truth table, then m i must be present in the minterm expansion. The minterm expansion for a function f is unique.  However, it is not necessarily the lowest cost.

6. Spring 2011ECE 301 - Digital Electronics6 Minterm Expansion The minterm expansion for a general function of 3 variables can be written as follows: Denotes the logical sum operation a i = 0 or 1. 3 variables This can be extended to n variables

7. Spring 2011ECE 301 - Digital Electronics7 Minterm Expansion: Example #1 Determine the minterm expansion for the function defined by the following truth table: ABCF 0000 0010 0100 0111 1000 1011 1101 1110

8. Spring 2011ECE 301 - Digital Electronics8 Minterm Expansion: Example #2 Determine the minterm expansion for each of the following Boolean expressions: F 1 (A,B,C) = A.B.C' + A.B'.C + A'.B'.C + A.B.C F 2 (A,B,C) = A.C' + A.B + B'.C

9. Spring 2011ECE 301 - Digital Electronics9 Maxterm In general, a maxterm of n variables is a sum (ORing) of n literals in which each variable appears exactly once in either true or complemented form, but not both.  A literal is a variable or its complement. For a given row in the truth table, the corresponding maxterm is formed by  Including the true form a variable if its value is 0.  Including the complemented form of a variable if its value is 1.

10. Spring 2011ECE 301 - Digital Electronics10 Maxterms

11. Spring 2011ECE 301 - Digital Electronics11 Maxterm Expansion When a function f is written as a product (ANDing) of maxterms, it is referred to as a maxterm expansion or a standard product of sums.  aka. “canonical product of sums”  aka. “conjunctive normal form” If f = 0 for row i of the truth table, then M i must be present in the maxterm expansion. The maxterm expansion for a function f is unique.  However, it is not necessarily the lowest cost.

12. Spring 2011ECE 301 - Digital Electronics12 Maxterm Expansion The maxterm expansion for a general function of 3 variables can be written as follows: Denotes the logical product operation a i = 0 or 1. 3 variables This can be extended to n variables

13. Spring 2011ECE 301 - Digital Electronics13 Maxterm Expansion: Example #1 Determine the maxterm expansion for the function defined by the following truth table: ABCF 0000 0010 0100 0111 1000 1011 1101 1110

14. Spring 2011ECE 301 - Digital Electronics14 Maxterm Expansion: Example #2 Determine the maxterm expansion for each of the following Boolean expressions: F 1 (A,B,C) = (A+B+C').(A+B'+C).(A'+B'+C).(A+B+C) F 2 (A,B,C) = (A+C').(A+B).(B'+C)

15. Spring 2011ECE 301 - Digital Electronics15 Minterm and Maxterm Expansions What is the relationship between the minterm expansion and maxterm expansion for the same function?

16. Spring 2011ECE 301 - Digital Electronics16 Minterm and Maxterm Expansions What is the relationship between the minterm expansion for a function and that for the complement of the function? What about the maxterm expansion?

17. Spring 2011ECE 301 - Digital Electronics17 Minterm and Maxterm Expansions

18. Spring 2011ECE 301 - Digital Electronics18 Logic Circuits A function f can be represented by either a minterm expansion or a maxterm expansion. Both forms of the function can be realized using logic gates that implement the basic logic operations. Minterm Expansion (Standard SOP)  Consists of the sum (OR) of product (AND) terms.  Realized using an AND-OR circuit. Maxterm Expansion (Standard POS)  Consists of the product (AND) of sum (OR) terms.  Realized using an OR-AND circuit.

19. Spring 2011ECE 301 - Digital Electronics19 Logic Circuits: Example For the function defined by the following truth table, 1. Determine the minterm expansion 2. Draw the circuit diagram ABCF 0000 0011 0101 0110 1001 1010 1101 1111

20. Spring 2011ECE 301 - Digital Electronics20 Logic Circuits: Example For the same function, 1. Determine the maxterm expansion 2. Draw the circuit diagram Which logic circuit is “cheaper”?

21. Spring 2011ECE 301 - Digital Electronics21 Incompletely Specified Functions

22. Spring 2011ECE 301 - Digital Electronics22 Incompletely Specified Functions A function f is completely specified when its output is defined (i.e. either 0 or 1) for all combinations of its inputs. However, if the output of a function f is not defined for all combinations of its inputs, then it is said to be incompletely specified.  Those combinations of the inputs for which the output of function f is not defined are referred to as “don't care” outputs.

23. Spring 2011ECE 301 - Digital Electronics23 Incompletely Specified Functions The truth table representing an incompletely specified function includes an “x” (or a “d”) in each row corresponding to an input combination for which the output is not defined. ABCF 0000 001X 0101 011X 1001 1010 110X 1111 “don't care” for ABC = 001 “don't care” for ABC = 011 “don't care” for ABC = 110

24. Spring 2011ECE 301 - Digital Electronics24 Incompletely Specified Functions ABCF 0000 001X 0101 011X 1001 1010 110X 1111 The minterm expansion is: The maxterm expansion is: F(A,B,C) =  m(2,4,7) +  d(1,3,6) F(A,B,C) =  M(0,5).  D(1,3,6) “don't care” minterms “don't care” maxterms A “don't care” can be either a 0 or 1.  Select a value for each “don't care” that will help simplify the function.

25. Spring 2011ECE 301 - Digital Electronics25 Incompletely Specified Functions ABCF 0000 001X1 0101 011X2 1001 1010 110X3 1111 Assume X1 = 0, X2 = 0, X3 = 0: Assume X1 = 1, X2 = 1, X3 = 1: F(A,B,C) = A'BC' + AB'C' + ABC F(A,B,C) = B + AC' + A'C Assume X1 = 0, X2 = 1, X3 = 1: F(A,B,C) = B + AC'

26. Spring 2011ECE 301 - Digital Electronics26 Questions?