meljun cortes combinational logic (part 1)

8/8/2019 MELJUN CORTES Combinational Logic (Part 1)

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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:


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


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:


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:


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


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 )


Logic Diagram for the F Product of Sums

meljun cortes combinational logic (part 1)

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