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Lecture No. 9. Computer Logic Design Boolean Algebra and Logic Simplification. Veracity of DeMorgan's Theorems. First Theorem Second Theorem Alternative Method – use Truth Tables. Application of DeMorgan's Theorems. Apply to any number of variables Apply to combination of variables. - PowerPoint PPT Presentation
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Lecture No. 9
Computer Logic Design
Boolean Algebra and Logic Simplification
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• First Theorem
• Second Theorem
• Alternative Method – use Truth Tables
Veracity of DeMorgan's Theorems
BABA . A
BB.A
A
BBA
BABA .A
BBA
A
BB.A
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Application of DeMorgan's Theorems
• Apply to any number of variables
• Apply to combination of variables
ZYXZ.Y.X Z.Y.XZYX
BCACBA ).().(
).().().).(.( BCACBABCACBA
BCACBA .).().(.
CBBACABA ....
CBCABA ...
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• Finding Complement of a Function• Example:
)ZY).(ZX).(YX(F
Z.Y.Z.X.Y.XF
Z.YZ.XY.XF
Z.YZ.XY.XF
Application of DeMorgan's Theorems
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Shortcut for finding Complement of a Function1. Take dual of the function
• swap 1’s and 0’s• Swap AND and OR gates• Helpful to add parenthesis
2. Complement each literal
)CBA).(CBA(F
)CBA()CBA(F
CBACBAF
Application of DeMorgan's Theorems
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Types of Boolean Expressions
• Define Domain of an expression– set of all variables (complemented or otherwise)
• Boolean expressions may be expressed as:– Sum-of-Products (SOP) Form– Product-of-Sums (POS) Form– Each form may contain single variable terms– May contain complemented and un-complemented
terms– A SOP and POS expression can’t have a term of more
than one variable having an over bar extending over the entire term
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Sum-of-Products (SOP) Form
• Two or more product terms summed by Boolean addition
• Any expression -> SOP using Boolean algebra
• Examples:* A + BC
DCBAADDCADBADCB*
CBA CBA ABC *
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Sum-of-Products (SOP) Form
BEFBCDAB)EFCD(BAB*
BDBCBADACAB)DCB)(BA(*
BADAC
CBCAC)BA(C)BA(C)BA(*
•Conversion to SOP Form:
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Implementation of SOP Expression
A
B
C
B+AC+AD
AD
B+AC+AD
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Standard SOP Form & Minterms• SOP expressions containing all Variables in the
Domain in each term are in Standard Form.• Standard product terms are also called Minterms.• Any non-standard SOP expression may be
converted to Standard form by applying Boolean Algebra Rule 6 to it.
• Example:
)1AA(
CBACABCBA
)BB(CACBA
CACBA
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Standard SOP Form
• Example: Determine Standard SOP expression
CBACBACABABCCBA
)CBCBCBBC(ACBA
)CC)(BB(ACBA
ACBA
SHORTCUT: Introduce all possible combinations of the missing variables AND’ed with the original term
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Characteristics of a Minterm
• Minterm is a standard product term in which all variables appear exactly once (complemented or uncomplemented)
• Represents exactly one combination of the binary variables in a truth table for which the function produces a “1” output. That is the binary representation or value.
• Has value of 1 for that combination and 0 for all others• For n variables, there are 2n distinct minterms• Example:
1010
DCBA
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Product-of-Sums (POS) Form
• Two or more sum terms multiplied by Boolean multiplication
• Any expression -> POS using Boolean algebra• Examples:
(A+B)(B+C)(A+B+C)
)ON)(ONL)(ML(
)ZYX)(ZY)(X(
)DCBD)(B(A )C(A
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Product-of-Sums (POS) Form
)DCA(B)DC(BAB*
)DB)(CB(A)CDB(AACDAB*
C)BA(C)BA(C)BA(*
•Conversion to POS Form:
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Implementation of POS expression
D
C
(A+B)(B+C+D)(A+C)
A
B
(A+B)(B+C+D)(A+C)
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Standard POS Form & Maxterms• POS expressions containing all Variables in the
Domain in each term are in Standard Form.• Standard sum terms are also called Maxterms. A
Maxterm is a NOT Minterm. • Any non-standard POS expression may be
converted to Standard form by applying Boolean Algebra Rule 8 and Rule 12 A+BC=(A+B)(A+C) to it.
)0A.A(
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Standard POS Form• Example:
{Rule 8}
{Rule 12} )CBA)(CBA)(CBA(
)BCA)(BCA)(CBA(
)BBCA)(CBA(
)CA)(CBA(
SHORTCUT: Introduce all possible combinations of the missing variables OR’ed with the original term
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Characteristics of a Maxterm
• Maxterm is a standard sum term in which all variables appear exactly once (complemented or uncomplemented)
• Represents exactly one combination of the binary variables in a truth table for which the function produces a “0” output. That is the binary representation or value.
• Has value of 0 for that combination and 1 for all others• For n variables, there are 2n distinct maxterms• Example:
)1100(
)DCBA(
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Why Standard SOP and POS Forms?
• Direct mapping of Standard Form expressions and Truth Table entries.
• Alternate Mapping methods for simplification of expressions
• Minimal Circuit implementation by switching between Standard SOP or POS
• PLD based function implementation
