ECE 301 – Digital Electronics

Minterm and Maxterm Expansionsand

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.

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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 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. 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:

Denotes the logical sum operation

ai = 0 or 1.

3 variables

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:

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 Including the true form a variable if its value is 0. Including the complemented form of a variable if

its value is 1.

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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. However, it is not necessarily the lowest cost.

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Maxterm Expansion The maxterm expansion for a general function

of 3 variables can be written as follows:

Denotes the logical product operation

ai = 0 or 1.

3 variables

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:

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) 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 expansion2. Draw the circuit diagram

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 1 1

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Logic Circuits: Example

For the same function,

1. Determine the maxterm expansion2. 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.

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.

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

“don't care” for ABC = 001

“don't care” for ABC = 011

“don't care” for ABC = 110

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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:

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.

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Incompletely Specified Functions

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

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'

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Questions?