Download Lecture #6 - the GMU ECE Department

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, 6th Edition, by Roth and Kinney,
and were used with permission from Cengage Learning.
Minterms and Maxterms
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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
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
Including the true form a variable if its value is 1.

Including the complemented form of a variable if
its value is 0.
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Minterms
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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 mi must be
present in the minterm expansion.
The minterm expansion for a function f is unique.

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However, it is not necessarily the lowest cost.
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Minterm Expansion

The minterm expansion for a general function
of 3 variables can be written as follows:
3 variables
Denotes the logical
sum operation
ai = 0 or 1.
This can be extended to n variables
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Minterm Expansion: Example #1
Determine the minterm expansion for the
function defined by the following truth table:
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A
B
C
F
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
1
1
1
0
1
1
1
1
0
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Minterm Expansion: Example #2
Determine the minterm expansion for each of
the following Boolean expressions:
F1(A,B,C) = A.B.C' + A.B'.C + A'.B'.C + A.B.C
F2(A,B,C) = A.C' + A.B + B'.C
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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
Spring 2011

Including the true form a variable if its value is 0.

Including the complemented form of a variable if
its value is 1.
ECE 301 - Digital Electronics
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Maxterms
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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 Mi must be
present in the maxterm expansion.
The maxterm expansion for a function f is unique.

Spring 2011
However, it is not necessarily the lowest cost.
ECE 301 - Digital Electronics
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Maxterm Expansion

The maxterm expansion for a general function
of 3 variables can be written as follows:
3 variables
Denotes the logical
product operation
ai = 0 or 1.
This can be extended to n variables
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Maxterm Expansion: Example #1
Determine the maxterm expansion for the
function defined by the following truth table:
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A
B
C
F
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
1
1
1
0
1
1
1
1
0
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Maxterm Expansion: Example #2
Determine the maxterm expansion for each of the
following Boolean expressions:
F1(A,B,C) = (A+B+C').(A+B'+C).(A'+B'+C).(A+B+C)
F2(A,B,C) = (A+C').(A+B).(B'+C)
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Minterm and Maxterm Expansions
What is the relationship between the minterm
expansion and maxterm expansion for the
same function?
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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?
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Minterm and Maxterm Expansions
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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)
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
Consists of the product (AND) of sum (OR) terms.

Realized using an OR-AND circuit.
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Logic Circuits: Example
For the function defined by the following truth table,
1. Determine the minterm expansion
2. Draw the circuit diagram
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A
B
C
F
0
0
0
0
0
0
1
1
0
1
0
1
0
1
1
0
1
0
0
1
1
0
1
0
1
1
0
1
1
1
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1
19
Logic Circuits: Example
For the same function,
1. Determine the maxterm expansion
2. Draw the circuit diagram
Which logic circuit is “cheaper”?
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Incompletely Specified Functions
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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.

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Those combinations of the inputs for which
the output of function f is not defined are
referred to as “don't care” outputs.
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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.
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A
B
C
F
0
0
0
0
0
0
1
X
0
1
0
1
0
1
1
X
1
0
0
1
1
0
1
0
1
1
0
X
1
1
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“don't care” for ABC = 001
“don't care” for ABC = 011
“don't care” for ABC = 110
23
Incompletely Specified Functions

A
B
C
F
0
0
0
0
0
0
1
X
0
1
0
1
0
1
1
X
1
0
0
1
1
0
1
0
1
1
0
X
1
1
1
1
The minterm expansion is:
F(A,B,C) = Sm(2,4,7) + Sd(1,3,6)
“don't care” minterms
The maxterm expansion is:
F(A,B,C) = P M(0,5) . P D(1,3,6)
“don't care” maxterms
A “don't care” can be either a 0 or 1.

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Select a value for each “don't care” that will help
simplify the function.
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Incompletely Specified Functions
Assume X1 = 0, X2 = 0, X3 = 0:
A
B
C
F
0
0
0
0
0
0
1
X1
0
1
0
1
0
1
1
X2
1
0
0
1
1
0
1
0
1
1
0
X3
1
1
1
1
F(A,B,C) = A'BC' + AB'C' + ABC
Assume X1 = 1, X2 = 1, X3 = 1:
F(A,B,C) = B + AC' + A'C
Assume X1 = 0, X2 = 1, X3 = 1:
F(A,B,C) = B + AC'
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Questions?
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