Combinational Logic, Spring 1999 1 TWO-LEVEL COMBINATIONAL LOGIC OVERVIEW Canonical forms Two-level simplification Boolean cubes Karnaugh maps Quine-McClusky (Tabulation) Method Don't care terms Canonical and Standard Forms Minterms and Maxterms For two binary variables and combined with an AND operation, the or E F minterms standard products are: , and . That is, two binary variables provide EF EF E F EF w w w w # % 8 # # 8 possible combinations (minterms.) variables have minterms. Each minterm has each variable being primed if the corresponding bit of the binary number is a and ! unprimed if a " Similarly, two binary variables and combined with an OR operation, the E F maxterms or are: , and . Two binary variables also standard sums E F E FE F E F w w w w provide possible combinations (maxterms) and variables have maxterms. Each # % 8 # # 8 maxterm has each variable being primed if the corresponding bit of the binary number is a " and unprimed if a ! A maxterm is the complement of its corresponding minterm, and vice versa. Sum of Products (or Minterms) A Boolean function can be expressed as a sum of minterms. The minterms whose sum defines the Boolean function are those that give the 1's of the function in a truth table. Product of Sums (or Maxterms) A Boolean function can be expressed as a product of maxterms. The maxterms whose sum defines the Boolean function are those that give the 0's of the function in a truth table. Minterms and Maxterms for Three Binary Variables
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Combinational Logic, Spring 1999
1
TWO-LEVEL COMBINATIONAL LOGIC
OVERVIEW
Canonical forms
Two-level simplification
Boolean cubes
Karnaugh maps
Quine-McClusky (Tabulation) Method
Don't care terms
Canonical and Standard Forms
Minterms and Maxterms
For two binary variables and combined with an AND operation, the orE F minterms
standard products are: , and . That is, two binary variables provideEFß EF ß E F E Fw w w w
# ú % 8 ## 8 possible combinations (minterms.) variables have minterms. Each minterm has
each variable being primed if the corresponding bit of the binary number is a and!
unprimed if a "Þ
Similarly, two binary variables and combined with an OR operation, the E F maxterms
or are: , and . Two binary variables alsostandard sums E Ä Fß E Ä F ß E Ä F E Ä Fw w w w
provide possible combinations (maxterms) and variables have maxterms. Each# ú % 8 ## 8
maxterm has each variable being primed if the corresponding bit of the binary number is a "
and unprimed if a !Þ
A maxterm is the complement of its corresponding minterm, and vice versa.
Sum of Products (or Minterms)
A Boolean function can be expressed as a sum of minterms. The minterms whose sum
defines the Boolean function are those that give the 1's of the function in a truth table.
Product of Sums (or Maxterms)
A Boolean function can be expressed as a product of maxterms. The maxterms whose
sum defines the Boolean function are those that give the 0's of the function in a truth table.
Minterms and Maxterms for Three Binary Variables
Combinational Logic, Spring 1999
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Minterms Maxterms
Term Designation Term Designation\ ] ^
! ! !
! ! "
! " !
! " "
" ! !
" ! "
" " !
" " "
\ ] ^ 7 \ Ä ]
\ ] ^ 7
\ ] ^ 7
\ ] ^ 7
\] ^ 7
\] ^ 7
\] ^ 7
\] ^ 7
w w w!
w w"
w w#
w$
w w%
w&
w'
(
Ä ^ Q
\ Ä ] Ä ^ Q
\ Ä ] Ä ^ Q
\ Ä ] Ä ^ Q
\ Ä ] Ä ^ Q
\ Ä ] Ä ^ Q
\ Ä ] Ä ^ Q
\ Ä ] Ä ^ Q
!
w"
w#
w w$
w%
w w&
w w'
w w w(
Example: Given a three-variable truth table as follows:
\ ] ^ J J
! ! ! ! !
! ! " " !
! " ! ! !
! " " ! "
" ! ! " !
" ! " ! "
" " ! ! "
" " " " "
Function Function " #
J J" #and can be expressed as a sum of products as follows:
Combinational Logic, Spring 1999
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J ú \ ] ^ Ä \] ^ Ä \] ^ ú 7 Ä 7 Ä 7" " % (w w w w
J ú \ ] ^ Ä \] ^ Ä \] ^ Ä \] ^ ú 7 Ä 7 Ä 7 Ä 7# $ & ' (w w w
J J" #and can also be expressed as a product of sums as follows:
J ú Ð\ Ä ] Ä ^ ÑÐ\ Ä ] Ä ^ ÑÐ\ Ä ] Ä ^ ÑÐ\ Ä ] Ä ^ ÑÐ\ Ä ] Ä ^ Ñ"w w w w w w w