CombinationalLogic 1
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Topics
• Basics of digital logic • Basic functions♦ Boolean algebra♦ Gates to implement Boolean functions
• Identities and Simplification
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Binary Logic
• Binary variables♦ Can be 0 or 1 (T or F, low or high)♦ Variables named with single letters in
examples♦ Use words when designing circuits
• Basic Functions♦ AND♦ OR♦ NOT
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AND Operator
• Symbol is dot♦ Z = X · Y
• Or no symbol♦ Z = XY
• Truth table ->• Z is 1 only if♦ Both X and Y are 1
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OR Operator
• Symbol is +♦ Not addition♦ Z = X + Y
• Truth table ->• Z is 1 if either 1♦ Or both!
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NOT Operator
• Unary• Symbol is bar (or ’)♦ Z = X’
• Truth table ->• Inversion
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Gates
• Circuit diagrams are traditionally used to document circuits
• Remember that 0 and 1 are represented by voltages
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AND Gate
Timing Diagrams
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OR Gate
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Inverter
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More Inputs
• Work same way• What’s output?
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Digital Circuit Representation: Schematic
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Digital Circuit Representation: Boolean Algebra
• For now equations with operators AND, OR, and NOT
• Can evaluate terms, then final OR
• Alternate representations next
ZY X F
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Digital Circuit Representation: Truth Table
• 2n rowswhere n # ofvariables
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Functions
• Can get same truth table with different functions
• Usually want simplest function♦ Fewest gates or using particular types
of gates♦ More on this later
ZY X F ))(( ZXY X F
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Identities
• Use identities to manipulate functions
• On previous slide, I used distributive law
to transform fromZY X F ))(( ZXY X F
))(( ZX YX YZ X
to
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Table of Identities
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Duals
• Left and right columns are duals
• Replace AND with OR, 0s with 1s
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Single Variable Identities
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Commutative
• Order independent
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Associative
• Independent of order in which we group
• So can also be written asand
ZYX XYZ
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Distributive
• Can substitute arbitrarily large algebraic expressions for the variables
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DeMorgan’s Theorem
• Used a lot• NOR equals invert AND
• NAND equals invert OR
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Truth Tables for DeMorgan’s
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Algebraic Manipulation
• Consider functionXZZYX YZ X F
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Simplify Function
XZZ Z YX F )(
XZYX F 1
XZYX F
Apply
Apply
Apply
XZZYX YZ X F
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Fewer GatesXZYX F
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Consensus Theorem
• The third term is redundant♦ Can just drop
• Proof in book, but in summary♦ For third term to be true, Y & Z both 1♦ Then one of the first two terms must be
1!
ZXXYYZZXXY
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Complement of a Function
• Definition:
1s & 0s swapped in truth table
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Truth Table of the Complement of a Function
X Y Z F = X + Y’Z F’
0 0 0 0 1
0 0 1 1 0
0 1 0 0 1
0 1 1 0 1
1 0 0 1 0
1 0 1 1 0
1 1 0 1 0
1 1 1 1 0
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Algebraic Form for Complement• Mechanical way to derive
algebraic form for the complement of a function
1. Take the dual• Recall: Interchange AND & OR, and 1s & 0s
2. Complement each literal (a literal is a variable complemented or not; e.g. x , x’ , y, y’ each is a literal)
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Example: Algebraic form for the complement of a function
F = X + Y’Z• To get the complement F’
1. Take dual of right hand side
X . (Y’ + Z)2. Complement each literal: X’ . (Y
+ Z’)
F’ = X’ . (Y + Z’)
Mechanically Go From Truth Table to Function
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From Truth Table to Function
• Consider a truth table• Can implement F
by taking OR of all terms that correspond to rows for which F is 1 “Standard Form” of
the function
XYZZXYZYXZY XZ YX F
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Standard Forms
• Not necessarily simplest F• But it’s mechanical way to go
from truth table to function
• Definitions:♦ Product terms – AND ĀBZ♦ Sum terms – OR X + Ā♦ This is logical product and sum, not
arithmetic
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Definition: Minterm
• Product term in which all variables appear once (complemented or not)
• For the variables X, Y and Z example minterms : X’Y’Z’, X’Y’Z, X’YZ’, …., XYZ
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Definition: Minterm (continued)
MinTerm
Each minterm represents exactly one combination of the binary variables in a truth table.
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Truth Tables of Minterms
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Number of Minterms
• For n variables, there will be 2n minterms
• Minterms are labeled from minterm 0, to minterm 2n-1
♦m0 , m1 , m2 , … , m2n
-2 , m2n
-1
• For n = 3, we have
♦m0 , m1 , m2 , m3 , m4 , m5 , m6 , m7
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Definition: Maxterm
• Sum term in which all variables appear once (complemented or not)
• For the variables X, Y and Z the maxterms are:
X+Y+Z , X+Y+Z’ …. , X’+Y’+Z’
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Definition: Maxterms (continued)
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm,mxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx,mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm
Maxterm
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Truth Tables of Maxterms
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Minterm related to Maxterm
• Minterms and maxterms with same subscripts are complements
• Example33 MZYXYZXm
Mjm j
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Standard Form of F:Sum of Minterms
• OR all of the minterms of truth table for which the functionvalue is 1
• F = m0 + m2 + m5 + m7
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Complement of F
• Not surprisingly, just sum of the other minterms
• In this caseF’ = m1 + m3 + m4 + m6
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Product of Maxterms
• Recall that maxterm is true except for its own row
• So M1 is only false for 001
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Product of Maxterms
• F = m0 + m2 + m5 + m7
• Remember:♦ M1 is only false for 001♦ M3 is only false for 011♦ M4 is only false for 100♦ M6 is only false for 110
• Can express F as AND of M1, M3, M4, M6
6431 MMMMF
))(( ZYXZYXF ))(( ZYXZYX
or
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Recap
• Working (so far) with AND, OR, and NOT
• Algebraic identities• Algebraic simplification• Minterms and maxterms• Can now synthesize function
from truth table