1. Computer Logic and Digital Design Chapter 2 Henry Hexmoor AND logic
input
output
AND Gate B AB
S1 . S2
S S path?
off off no
on off no
off on no
on on yes
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2. OR logic input S1 + S2 S1 S2 path? off off no on off yes off on yes S1
on on yes S1 S2
output
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3. NOT logic S S' S S
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4. Equivalent Symbols of NAND, NOR Gates
NAND Symbols
Normal Symbol
Alternate NAND Symbol X Y
(X . Y)’ X Y
X’ + Y’
According to DeMorgan’s theorem T13: (X . Y)’ = X’ + Y’
NOR Symbols
Normal NOR Symbol
Alternate NOR Symbol X Y
(X + Y)’ X Y
X’ . Y’
According to DeMorgan’s theorem T13’: (X + Y)’ = X’ . Y’
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5. Boolean Algebra number of truth table rows = 2n
Literals (n) and terms
number of truth table rows = 2n
X Y Z Y’Z X+Y’Z
See table 2-3
Using demorgan’s laws OR and AND are interchanged… the duality principle
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6. Switching Algebra Axioms & Theorems
(A1) X = 0 if X ≠ 1 (A1’) X = 1 if X ≠ 0
(A2) If X = 0, then X’ = 1 (A2’) if X = 1, then, X’ = 0
(A3) = (A3’) = 1
(A4) = 1 (A4’) = 0
(A5) = = 0 (A5’) = = 1
(T1) X + 0 = X (T1’) X . 1 = X (Identities)
(T2) X + 1 = (T2’) X . 0 = (Null elements)
(T3) X + X = X (T3’) X . X = X (Idempotency)
(T4) (X’)’ = X (Involution)
(T5) X + X’ = 1 (T5’) X . X’ = (Complements)
(T6) X + Y = Y + X (T6’) X . Y = Y . X (Commutativity)
(T7) (X + Y) + Z = X + (Y + Z) (T7’) (X . Y) . Z = X . (Y . Z) (Associativity)
(T8) X . Y + X . Z = X . (Y + Z) (T8’) (X + Y) . (X + Z) = X + Y . Z (Distributivity)
(T9) X + X . Y = X (T9’) X . (X + Y) = X (Covering)
(T10) X . Y + X . Y’ = X (T10’) (X + Y) . (X + Y’) = X (Combining)
(T11) X . Y + X’. Z + Y . Z = X . Y + X’ . Z
(T11’) (X + Y) . ( X’ + Z) . (Y + Z) = (X + Y) . (X’ + Z) (Consensus)
(T12) X + X X = X (T12’) X . X X = X (Generalized idempotency)
(T13) (X1 . X Xn)’ = X1’ + X2’ Xn’
(T13’) (X1 + X Xn)’ = X1’ . X2’ Xn’ (DeMorgan’s theorems)
(T14) [F(X1, X2, . . ., Xn, +, .)]’ = F(X1’, X2’, . . ., Xn’, . , +) (Generalized DeMorgran’s theorem)
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7. Switching Algebra Axioms
First two axioms state that a variable X can only take on only one of two values:
(A1) X = 0 if X ¹ 1
(A1’) X = 1 if X ¹ 0
Not Axioms, formally define X’ (X prime or NOT X):
(A2) If X = 0, then X’ = 1
(A2’) if X = 1, then, X’ = 0
Note: Above axioms are stated in pairs with only difference
being the interchange of the symbols 0 and 1.
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8. Three More Switching Algebra Axioms
The following three Boolean Algebra axioms state and formally define the AND, OR operations:
(A3) = 0
(A3’) = 1
(A4) = 1
(A4’) = 0
(A5) = 1 .0 = 0
(A5’) = = 1
Axioms A1-A5, A1’-A5’ completely define switching algebra.
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9. Switching Algebra: Single-Variables Theorems
Switching-algebra theorems are statements known to be always true (proven using axioms) that allow us to manipulate algebraic logic expressions to allow for simpler analysis.
(e.g . X + 0 = X allow us to replace every X +0 with X)
The Theorems: (T1-T5, T1’-T5’)
(T1) X + 0 = X (T1’) X . 1 = X (Identities)
(T2) X + 1 = (T2’) X . 0 = 0 (Null elements)
(T3) X + X = X (T3’) X . X = X (Idempotency)
(T4) (X’)’ = X (Involution)
(T5) X + X’ = (T5’) X . X’ = 0 (Complements)
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10. Perfect Induction
Most theorems in switching algebra are simple to prove using perfect induction:
Since a switching variable can only take the values 0 and 1 we can prove a theorem involving a single variable X by proving it true for X = 0 and X =1
Example: To prove (T1) X + 0 = X
[X = 0] = 0 true according to axiom A4’
[X = 1] = 1 true according to axiom A5’
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11. Switching Algebra: Two- and Three-Variable Theorems
(Commutativity)
(T6) X + Y = Y + X
(T6’) X . Y = Y . X
(Associativity)
(T7) (X + Y) + Z = X + (Y + Z)
(T7’) (X . Y) . Z = X . (Y . Z)
T6-T7, T6’ -T7’ are similar to commutative and associative laws for
addition and multiplication of integers and reals.
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12. Two- and Three-Variable Theorems (Continued)
(Distributivity)
(T8) X . Y + X . Z = X . (Y + Z)
(T8’) (X + Y) . (X + Z) = X + Y . Z
T8 allows to multiply-out an expression to get sum-of-products form (distribute logical multiplication over logical addition):
For example:
V . (W + X) . (Y + Z) = V .W . Y + V. W. Z + V. X . Y + V. X . Z
sum-of-products form
T8’ allows to add-out an expression to get a product-of-sums form (distribute logical addition over logical multiplication):
(V . W . X) + (Y . Z ) = (V + Y) . (V + Z) . (W + Y) . (W + Z) . (X + Y) . (X + Z)
product-of-sums form
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13. Theorem Proof using Truth Table
Can use truth table to prove T8 by perfect induction.
i.e, Prove that: X . Y + X . Z = X . (Y + Z)
(i) Construct truth table for both sides of above equality. x y z
y + z
x.(y + z)
x.y
x.z
x.y + 1
(ii) Check that from truth table check that that X . Y + X . Z = X . (Y + Z)
This is satisfied because output column values for X . Y + X . Z and output column values for X . (Y + Z) are equal for all cases.
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14. Two- and Three-Variable Theorems (Continued)
(Covering)
(T9) X + X . Y = X
(T9’) X . (X + Y) = X
(Combining)
(T10) X . Y + X . Y’ = X
(T10’) (X + Y) . (X + Y’) = X
T9-T10 used in the minimization of logic functions.
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15. Two- and Three-Variable Theorems (Continued)
(Consensus)
(T11) X . Y + X’. Z + Y . Z = X . Y + X’ . Z
(T11’) (X + Y) . ( X’ + Z) . (Y + Z) = (X + Y) . (X’ + Z)
In T11 the term Y. Z is called the consensus of the term X . Y and the term X’ . Z:
If Y . Z = 1, then either X . Y or X’ . Z must also be 1.
Thus the term Y . Z is redundant and may be dropped.
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16. n-Variable Theorems (T13), (T13’) are probably the most commonly
(Generalized idempotency)
(T12) X + X X = X
(T12’) X . X X = X
(DeMorgan’s theorems)
(T13) (X1 . X Xn)’ = X1’ + X2’ Xn’
(T13’) (X1 + X Xn)’ = X1’ . X2’ Xn’
(T13), (T13’) are probably the most commonly
used theorems of switching algebra.
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17. Examples Using DeMorgan’s theorems
Example: Equivalence of NAND Gate:
A two-input NAND Gate has the output expression Z = (X . Y)’ using (T13) Z = (X . Y)’ = (X’ + Y’)
The function of a NAND gate can be achieved with an OR gate with an inverter at each input.
Example: Equivalence of NOR Gate
A two-input NOR Gate has the output expression Z=(X+Y)’
using (T13’) Z = (X + Y)’ = X’ . Y’
The function of a NOR gate can be achieved with an AND gate with an inverter at each input.
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18. n-Variable Theorems (Continued)
(Generalized DeMorgran’s theorem)
(T14) [F(X1, X2, . . ., Xn, +, .)]’ = F(X1’, X2’, . . ., Xn’, . , +)
States that given any n-variable logic expression its complement can be found by swapping + and . and complementing all variables.
Example:
F(W,X,Y,Z) = (W’.X) + ( X.Y) + (W.(X’ + Z’))
= ((W)’ . X) + (X. Y) + (W.((X)’ + (Z)’))
[F(W,X,Y,Z)]’ = ((W’)’ + X’) .(X’ + Y’).(W’ + ((X’)’.(Z’)’))
Using T4, (X’)’ = X simplifies it to:
[F(W,X,Y,Z)]’ = (W + X’) . (X’ + Y’) . (W’ + (X . Z))
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19. Logic Function Representation Definitions
A literal: is a variable or a complement of a variable
Examples: X, Y, X’, Y’
A product term: is a single literal, or a product of two or more literals. Examples: Z’ W.Y.Y X.Y’.Z W’.Y’.Z
A sum-of-products expression: is a logical sum of product terms.
Example: Z’ + W.X.Y + X.Y’.Z + W’.Y’.Z
A sum term: is a single literal or logical sum of two or more literals
Examples: Z’ W + X + Y X + Y’ + Z W’ + Y’ + Z
A product-of-sums expression: is a logical product of sum terms.
Example: Z’. (W + X + Y) . (X + Y’ + Z) . (W’ + Y’ + Z)
A normal term: is a product or sum term in which no variable appears more than once
Examples of non-normal terms: W.X.X.Y’ W+W+X’+Y X.X’.Y
Examples of normal terms: W . X . Y’ W + X’ + Y
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20. Logic Expression Algebraic Manipulation Example
Prove that the following identity is true using Algebraic expression Manipulation : (one can also prove it using a truth table)
X .Y + X . Z = ((X’ + Y’) . (X’ + Z’))’
Starting from the left hand side of the identity:
Let F = X .Y + X . Z
A = X . Y B = X . Z
Then F = A + B
Using DeMorgan’s theorem T 13 on F:
F = A + B = (A’ . B’)’ (1)
Using DeMorgan’s theorem T 13’ on A, B:
A = X . Y = (X’ + Y’)’ (2)
B = X . Z = (X’ + Z’)’ (3)
Substituting A, B from (2), (3), back in F in (1) gives:
F = (A’ . B’)’ = ((X’ + Y’) . (X’ + Z’))’
Which is equal to the right hand side of the identity.
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21. Terminology: Minterms
A minterm is a special product of literals, in which each input variable appears exactly once.
A function with n variables has 2n minterms (since each variable can appear complemented or not)
A three-variable function, such as f(x,y,z), has 23 = 8 minterms:
Each minterm is true for exactly one combination of inputs:
x’y’z’ x’y’z x’yz’ x’yz
xy’z’ xy’z xyz’ xyz
Minterm Is true when… Shorthand
x’y’z’ x=0, y=0, z=0 m0
x’y’z x=0, y=0, z=1 m1
x’yz’ x=0, y=1, z=0 m2
x’yz x=0, y=1, z=1 m3
xy’z’ x=1, y=0, z=0 m4
xy’z x=1, y=0, z=1 m5
xyz’ x=1, y=1, z=0 m6
xyz x=1, y=1, z=1 m7
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22. Terminology: Maxterms
A maxterm is a special sum of literals, in which each input variable appears exactly once.
A function with n variables has 2n maxterms (since each variable can appear complemented or not)
A three-variable function, such as f(x,y,z), has 23 = 8 maxterms:
Maxterm Shorthand
x’+ y’+ z’ M0
x’+ y’ + z M1
x’+ y + z’ M2
x’ + y + z M3
x + y’ + z’ M4
x + y’ + z M5
x + y + z’ M6
x + y + z M7
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23. Canonical Forms Minterms and Maxterms
Index Representation of Minterms and Maxterms
Sum-of-Minterm (SOM) Representations
Product-of-Maxterm (POM) Representations
Representation of Complements of Functions
Conversions between Representations
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24. Logic Function Representation Definitions
Minterm
An n-variable minterm is a normal product term with n literals.
There are 2n such products terms.
Example of 4-variable minterms:
W.X’.Y’.Z’ W.X.Y’.Z W’.X’.Y.Z’
Maxterm
An n-variable maxterm is a normal sum term with n literals.
There are 2n such sum terms.
Examples of 4-variable maxterms:
W’ + X’ + Y + Z’ W + X’ + Y’ + Z W’ + X’ + Y + Z
A minterm can be defined as as product term that is 1 in exactly one row of the truth table.
A maxterm can similarly be defined as a sum term that is 0 in exactly one row in the truth table.
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25. Minterms/Maxterms for A 3-variable function F(X,Y,Z)
Row X Y Z F Minterm Maxterm
F(0,0,0) X’.Y’.Z’ X + Y + Z
F(0,0,1) X’.Y’.Z X + Y + Z’
F(0,1,0) X’.Y.Z’ X + Y’ + Z
F(0,1,1) X’.Y.Z X + Y’ + Z’
F(1,0,0) X.Y’.Z’ X’ + Y + Z
F(1,0,1) X.Y’.Z X’ + Y + Z’
F(1,1,0) X.Y.Z’ X’ + Y’ + Z
F(1,1,1) X.Y.Z X’ + Y’ + Z’
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26. Canonical Sum Example The function represented by the truth table:
has the canonical sum representation:
F = S X,Y,Z m(0, 3, 4, 6, 7)
= X’.Y’.Z’ + X’.Y.Z + X.Y’.Z’ + X.Y’.Z’ + X.Y.Z
Row X Y Z F
Minterm list using S notation
Algebraic canonical sum of minterms
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27. Canonical Product Example
The function represented by the truth table:
has the canonical product representation:
F = P X,Y,Z M(1,2,5)
= (X + Y + Z’) . (X + Y’ + Z) . (X’ + Y + Z’)
Row X Y Z F
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28. Minterm and Maxterm Relationship
Review: DeMorgan's Theorem
and
Two-variable example:
M2 is the complement of m2 and vice-versa.
Since DeMorgan's Theorem holds for n variables, the above holds for terms of n variables
giving:
Thus Mi is the complement of mi. x y = x + y x + y = x × y M = x + y m = x· y 2 2 i m M =
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29. Conversion Between Minterm/Maxterm Lists
To convert between a minterm list and a maxterm list
take the set complement.
Example (page 44, 3rd ed.):
F’(X,Y, Z) = S m(1,3, 4, 6) = m1 + m3 + m4 + m6
F(X,Y, Z) = P M(1,3,4,6) = M1M3M4M6
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30. Verbal Synthesis Example: An Alarm Circuit
A verbal logic description:
The ALARM output is 1 if the panic input is 1, or if the ENABLE input is 1, the EXISTING input is 0, and the house is not secure.
The house is secure if the WINDOW, DOOR, GARAGE inputs are all 1
This can be put in logic expressions as follows:
ALARM = PANIC + ENABLE . EXISTING’ . SECURE’
SECURE = WINDOW. DOOR. GARAGE
ALARM = PANIC + ENABLE . EXISTING’. (WINDOW . DOOR . GARAGE)’
In sum of products form as (by using DeMorgan T13 and multiplying out) :
ALARM = PANIC + ENABLE. EXISTING’ . WINDOW’
+ ENABLE . EXISTING’. DOOR’+ ENABLE. EXISTING’. GARAGE’
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31. cost criteria 2-4 OR Gate Literal cost
Gate input cost: literals + terms
OR Gate
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32. Combinational Circuit Minimization
Canonical sum and product logic expressions do not provide a circuit realization with the minimum number of gates.
Minimization methods reduce the cost of two level AND-OR, NAND-NAND, OR-AND, NOR-NOR circuits in three ways:
By minimizing the number of first level gates
By minimizing the number of inputs of each first-level gate.
Minimizing the inputs of the second level gate
Most minimization methods are based on the combining theorems T10, T10’:
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33. Karnaugh Maps
A Karnaugh Map or (K-map for short) is a graphical representation of the truth table of a logic function.
The K-map for an n-input logic function is an array with 2n cells or squares, one for each input combination or minterm.
The rows and columns are labeled so that the input combination for any cell is determined from the row and column headings.
The row and columns of the map are ordered in such a way that each cell differs from an adjacent cell in only one input variable:
Thus for an n-variable K-map, each cell has n adjacent cells.
The K-map for a function is filled by putting:
a ‘1’ in the square corresponding to a minterm
a ‘0’ otherwise (maybe omitted)
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34. 2-Variable K-map Truth Table: K-map Truth Table: K-map
For a 2-variable logic function F(X,Y):
Truth Table: X Y 1 2 3
Row X Y F Minterm
F(0,0) X’.Y’
F(0,1) X’.Y
F(1,0) X.Y’
F(1,1) X .Y
K-map
Example: For the function F(X,Y) = S X,Y m(1,2,3)
Truth Table:
K-map X Y 1 2 3
Row X Y F 1 1 1
Henry Hexmoor

35. 3 variable Karnaugh map Textbook conventionYZ 1 X
X’ Y Z’
X’ Y Z
X’ Y Z
X’ Y Z
XY’ Z’
X Y’Z
X Y Z
X Y Z’
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36. 3-Variable K-map For a 3-variable logic function F(X,Y,Z):
Truth Table:
K-map
Row X Y Z F Minterm
F(0,0,0) X’.Y’.Z’
F(0,0,1) X’.Y’.Z
F(0,1,0) X’.Y.Z’
F(0,1,1) X’.Y.Z
F(1,0,0) X.Y’.Z’
F(1,0,1) X.Y’.Z
F(1,1,0) X.Y.Z’
F(1,1,1) X.Y.Z X Y 1 YZ 2 3 6 7 4 5
Example: For the function F(X,Y,Z) = S X,Y,Z (1,4,6,7) X Y 1 Z YZ 2 3 6 7 4 5
K-map
Row X Y Z F 1
Truth Table: 1 1 1
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37. 4-Variable K-map Truth Table: K-map
For a 4-variable logic function F(W,X,Y,Z):
Truth Table:
K-map W Y 00 01 11 10 Z WX YZ X 1 3 2 4 5 7 6 12 13 15 14 8 9
Row W X Y Z F Minterm
F(0,0,0,0) W’.X’.Y’.Z’
F(0,0,0,1) W’. X’.Y’.Z
F(0,0,1,0) W’. X’.Y.Z’
F(0,0,1,1) W’. X’.Y.Z
F(0,1,0,0) W’. X.Y’.Z’
F(0,1,0,1) W’.X.Y’.Z
F(0,1,1,0) W’.X.Y.Z’
F(0,1,1,1) W’.X.Y.Z
F(1,0,0,0) W.X’.Y’.Z’
F(1,0,0,1) W.X’.Y’.Z
F(1,0,1,0) W.X’.Y.Z’
F(1,0,1,1) W.X’.Y.Z
F(1,1,0,0) W.X.Y’.Z’
F(1,1,0,1) W.X.Y’.Z
F(1,1,1,0) W.X.Y.Z’
F(1,1,1,1) W.X.Y.Z
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38. Minimization Using K-maps
Group or combine as many adjacent 1-cells as possible:
The larger the group is, the fewer the number of literals in the resulting product term.
Grouping 2 adjacent 1-cells eliminates 1 variable, grouping cells eliminates 2 variables, grouping 8 1-cells eliminates 3 variables, and so on. In general, grouping 2n squares eliminates n variables.
Select as few groups as possible to cover all the 1-cells (minterms) of the function:
The fewer the groups, the fewer the number of product terms in the minimized function.
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39. Karnaugh maps Example 2-4 (Figure 2-14b, page 54, 3rd ed.)
000
100
010
110
1cell Square overlap is OK. Z’ YZ 1 X 1 1 1 1 1
XY’
100
101
F2(X,Y,Z) = m(0,2,4,5,6) = Z’ + XY’
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40. 4 variable Karnaugh map example 2-5 (Figure 2-19, page 57, 3rd ed.)YZ
0000
0001
1100
1101
1000
1001
0000
0001
0100
0101
0010
0110 00 01 11 10 1
0100
1100
0110
1110 1 1 1 1 WX 1 1 1 1 1 1
F(W,X,Y,Z) = S m(0,1,2,4,5,5,8,9,12,13,14) = Y’ + W’Z’ + XZ’
Henry Hexmoor

41. prime/essential implicants
W’Z’ and XZ’ are prime implicants
(each rectangle needs all its cells)
Y’ is an essential prime implicant…
(the rectangle contains exclusive (i.e., nonshared) cells YZ 00 01 11 10 1 1 1 1 1 WX 1 1 1 1 1 1
F(X,Y,Z) = m(0,1,2,4,5,5,8,9,12,13,14) = Y’ + W’Z’ + XZ’
Selection rule: minimize overlap among prime implicants
Henry Hexmoor

42. product of sums: MaxtermsYZ 00 01 11 10 1 1 1 1 1 WX 1 1 1 1 1 1
F(X,Y,Z) = m(0,1,2,4,5,5,8,9,12,13,14) = YZ + WX’Y
 (Y’ + Z’) (W’ + X + Y)
Selection rule: minimize overlap among prime implicants
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43. More Examples…
x’z + y’z + xyz’.
x’z’ + xy’z
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44. multilevel optimization 2-6
Manipulate equations algebraically
G = AC’E + AC’F + AD’E + AD’F + BCDE’F’
= A (C’E + C’F + D’E + D’F) + BCDE’F’
= A (C’ + D’)(E + F) + BCDE’F’
OR Gate
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45. other gate types OR Gate See figures 2-22 and 2-23 in 4th edition
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46. HW 2
Demonstrate by means of truth tables the validity of the following identity:
(XYZ)’ = X’ + Y’ + Z’
(Q 2-1)
2. Optimize the following Boolean function by means a three-variable map:
F(X,Y,Z) = Sm(1,3,6,7)
(Q 2-14)
OR Gate
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