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a Boolean expression that have binary variablesand operators plus a binary variable as an
BooleanFunction
output
F = X + Y’Z
is evaluated by determining the binary value ofthe expression for all possible combinations ofthe binary input variables
Input variables
Output variable
Combinational Logic (Part 1) * Property of STI Page 1 of 12
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Example 1:
Theorem 6b is a form of absor tion where the
BooleanFunction
Boolean function F is F = X(X +Y) and isequivalent to F = X . More precisely, theexpression holds
X (X +Y )=X
Prove the expression above and illustrate thedifferent ways of writing down F in algebraic
form. How many ways can we represent F using a truth table?
Solution:
Combinational Logic (Part 1) * Property of STI Page 2 of 12
Step 1 F =XX +XY DistributiveProperty(Postulate 5b)
Step 2 F =X +XY Theorem 1b
Step 3 F =X (1)+XY Identity
Step 4 F =X (1+Y ) Distributive Property
Step 5 F =X 1) Theorem 2a
Step 6 F =X Identity
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Truth Table for F = X(X+Y)
BooleanFunction
Truth Table for F = X +XY
Combinational Logic (Part 1) * Property of STI Page 3 of 12
Hence,
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enable the process of simplification of Booleanexpressions digital logic circuits
StandardCanonical Forms
contain product and sum terms which imply thelogical operations AND and OR, respectively
Product terms
ANDed literals (ABCDE, XY’Z)
Sum terms
ORed literals (A+B +C +D +E, X +Y’ +Z)
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Minterm or Standard Product
re resents the least combination of the in ut
Minterms
variables for each output state in a truth table there are 2N minterms for N input variables
corresponding to each combination of inputvariables in a truth table
Example:
input variables: X , Y
22 = 4 minterms: X ’Y ’, X ’Y , XY ’, and XY
Minterms for Two Variables
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use only sum terms that contain all the inputvariables in either normal or complemented
Maxterms
form the symbol for a maxterm: M j where j is the
decimal value of its logical combination
The maxterms and minterms
with the same subscripts
are just the complements of each other.
M 3 = X’ + Y’ = X’Y’ = m 3’
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any Boolean function F may be expressed aseither a sum of minterms or a product of
CanonicalTerms
maxterms
a sum of minterms for which the function is 1:
F = X ’Y + XY ’ = m 1 + m 2
a product of maxterms for which the function is
equal to 0:
Combinational Logic (Part 1) * Property of STI Page 7 of 12
F = (X + Y) · X ’ + Y ’ = M 0 · M 3
the abbreviation of the expressions above:
F = Σ ΣΣ Σ m (1,2) = Π ΠΠ Π M (0,3)
and
F’ = Σ ΣΣ Σ m (0,3) = Π ΠΠ Π M (1,2)
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Example 2:
CanonicalTerms
Obtain the value of the function:
F = Σ ΣΣ Σ m (0,1,2,3,4,5,6,7)
Solution:
Minterms and Maxterms of Three Variables
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Solution (cont.):
CanonicalTerms
a function that includes all its minterms willalways lead to a value of 1
F = ΣΣΣΣ m (0,1,2,3,4,5,6,7) = 1
the product term that includes all maxterms will
always be equal to 0F = Π ΠΠ Π M (0,1,2,3,4,5,6,7) = 0
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Example 3:
Obtain the sum of minterms expression and the
CanonicalTerms
product of maxterms expression of the three-variable function
F = X + X ’Z + Y ’Z
Solution:
Combinational Logic (Part 1) * Property of STI Page 10 of 12
minterms of the function
F = Σ ΣΣ Σ m (1,2,3,4,5,6,7)
complement: F ’ = S m (0,2)
product of maxterms
F ’ = Π ΠΠ Π M (0,2)
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sum of products and the product of sums
StandardForms
Example:
Consider the Boolean function expressed as asum of products
F = B ’ + A’BC ’ + AB
Combinational Logic (Part 1) * Property of STI Page 11 of 12
Logic Diagram for the F Sum of Products
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Example:
Consider the Boolean function ex ressed as the
StandardForms
product of sumsF = B ’(A’ + B + C ’)(A + B )
Combinational Logic (Part 1) * Property of STI Page 12 of 12
Logic Diagram for the F Product of Sums