Digital Engineering
Ch6. Applications of Boolean
Algebra Minterm and Maxterm
Expansions
Junsung Choi
Assistant Professor
College of Electrical & Computer Engineering
4.2 Combinational Logic Design Using a Truth Table
Logic Design with Truth Tables
❖A switching circuit has three inputs and one output
⚫The inputs A, B, and C represent the first, second, and third bits, respectively, of a
binary number N
⚫The output of the circuit is to be f = 1 if N ≥ 0112 and f = 0 if N < 0112
❖An algebraic expression for f from the truth table by using the combinations
of values of A, B, and C for which f = 1
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4.2 Combinational Logic Design Using a Truth Table
Logic Design with Truth Table
❖Instead of writing f in terms of the 1’s of the function, we may also write f in
terms of the 0’s of the function
⚫Another way to derive Equation (4-3) is to first write f ′ as a sum of products, and
then complement the result: f ′ = A′B′C′ + A′B′C + A′BC′
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4.3 Minterm and Maxterm Expansions
Definition of Minterm and Maxterm
❖Minterm: A minterm of n variables is a product 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
❖Maxterm: A maxterm of n variables is a sum of n literals in which each
variable appears exactly once in either true or complemented form, but not
both
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4.3 Minterm and Maxterm Expansions
Minterm
❖A minterm of n variables is a product of n literals in which each variable
appears exactly once in either true or complemented form, but not both
❖Each minterm has a value of 1 for exactly one combination of values of the
variables A, B, and C
⚫Examples: If A = B = C = 0, A′B′C′ = 1; if A = B = 0 and C = 1, A′B′C = 1
❖Minterms are often written in abbreviated form – A′B′C′ is designated m0,
A′B′C is designated m1, etc
⚫In general, the minterm which corresponds to row i of the truth table is designated
mi (i is usually written in decimal)
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4.3 Minterm and Maxterm Expansions
Minterm
❖When a function f is written as a sum of minterms as in Equation (4-1), this is
referred to as a minterm expansion or a standard sum of products
⚫If f = 1 for row i of the truth table, then mi must be present in the minterm expansion
because mi = 1 only for the combination of values of the variables corresponding to
row i of the table
⚫Because the minterms present in f are in one-to-one correspondence with the 1’s of f
in the truth table, the minterm expansion for a function f is unique
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4.3 Minterm and Maxterm Expansions
Maxterm
❖A maxterm of n variables is a sum of n literals in which each variable appears
exactly once in either true or complemented form, but not both
❖Each maxterm has a value of 0 for exactly one combination of values for A, B,
and C
⚫Examples: If A = B = C = 0, A + B + C = 0; if A = B = 0 and C = 1, A + B + C′ = 0
❖Maxterms are often written in abbreviated form using M-notation
⚫The maxterm which corresponds to row i of the truth table is designated Mi
⚫Note that that each maxterm is the complement of the corresponding minterm, that
is, Mi = m′i
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4.3 Minterm and Maxterm Expansions
Maxterm
❖When a function f is written as a product of maxterms, as in Equation (4-3),
this is referred to as a maxterm expansion or standard product of sums
⚫If f = 0 for row i of the truth table, then Mi must be present in the maxterm
expansion because Mi = 0 only for the combination of values of the variables
corresponding to row i of the table
⚫Note that the maxterms are multiplied together so that if any one of them is 0, f will
be 0
⚫Because the maxterms are in one-to-one correspondence with the 0’s of f in the truth
table, the maxterm expansion for a function f is unique
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4.3 Minterm and Maxterm Expansions
Examples
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4.3 Minterm and Maxterm Expansions
Examples
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4.4 General Minterm and Maxterm Expansions
Example : Three Variables (A, B, C) case
❖In a truth table, each ai is a constant with a value 0 or 1
⚫To completely specify a function, we must assign values to all of the ai’s
❖Because each ai can be specified in two ways, there are 28 ways of filling the
F column of the truth table; therefore, there are 256 different functions of
three variables (This includes the degenerate cases, F identically equal to 0
and F identically equal to 1)
❖Minterm expansion for a general function of three variables:
❖Maxterm expansion for a general function of three variables:
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4.4 General Minterm and Maxterm Expansions
General Definition
❖For n variables,
❖Given two different minterms of n variables, mi and mj, at least one variable
appears complemented in one of the minterms and uncomplemented in the
other → i ≠ j, mimj = 0
⚫Example: For n = 3, m1m3 = (A′B′C)(A′BC) = 0
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4.4 General Minterm and Maxterm Expansions
General Definition
❖Given minterm expansions for two functions,
the product is
⚫Note that all of the cross-product terms (i ≠ j) drop out so that f1f2 contains only
those minterms which are present in both f1 and f2
⚫Example
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4.4 General Minterm and Maxterm Expansions
Conversion of Forms
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4.5 Incompletely Specified Functions
Example
❖Consider the following example in which the output of circuit N1 drives the
input of circuit N2
⚫The output of N1 does not generate all possible combinations of values for A, B, and
C
⚫In particular, we will assume that there are no combinations of values for w, x, y,
and z which cause A, B, and C to assume values of 001 or 110
⚫Hence, when we design N2, it is not necessary to specify values of F for ABC = 001
or 110 because these combinations of values can never occur as inputs to N2
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4.5 Incompletely Specified Functions
Example
❖The X’s in the table indicate that we don’t care whether the value of 0 or 1 is
assigned to F for the combinations ABC = 001 or 110
⚫In this example, we don’t care what the value of F is because these input
combinations never occur anyway
❖The function F is then incompletely specified
❖The minterms A′B′C and ABC ′ are referred to as don’t-care minterms, since
we don’t care whether they are present in the function or not
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4.5 Incompletely Specified Functions
Example
❖When we realize the function, we must specify values for the don’t-cares
→ It is desirable to choose values which will help simplify the function
⚫Example
⚫If we assign the value 0 to both X’s,
⚫If we assign 1 to the first X and 0 to the second,
⚫If we assign 1 to both X’s,
⚫Note that the second choice of values leads to the simplest solution
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4.5 Incompletely Specified Functions
Example
❖Notation
⚫When writing the minterm (maxterm) expansion for an incompletely specified
function, we will use m (M) to denote the required minterms (maxterms) and d (D)
to denote the don’t-care minterms (maxterms)
 F =  m ( 0, 3, 7 ) +  d (1, 6 ) or F =  M ( 2, 4, 5 )   D (1, 6 )
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