CHAPTER 3

Boolean Algebra
Contents









Describing Logic Circuits Algebraically
rules of evaluating a Boolean expression
Boolean Theorem...
Describing Logic Circuits Algebraically


OR gate, AND gate, and NOT circuit are
the basic building blocks of digital sys...
Circuits containing Inverters

4
Evaluating Logic Circuit Outputs


Let A=0, B=0, C=1, D=1, E=1
X

= [D+ ((A+B)C)'] • E
= [1 + ((0+0)1 )'] • 1
= [1 + (0•1...
Rules of evaluating a Boolean expression
First, perform all inversions of single
terms; that is, 0 = 1 or 1 = 0.
 Then pe...
Determining Output Level from a Diagram

7
Boolean
Theorems

8
Multivariable Theorems
(9)
(10)
(11)
(12)
(13.a)
(13.b)
(13.c)
(14)
(15)
(16)

x + y = y + x (Commutative law)
x • y = y •...
Proof of (14, 15, 16)
x + xy

x + x’y

= x (1+y)
= x • 1 [using theorem (6)]
= x [using theorem (2)]
= ( x + x’) (x + y) [...
DeMorgan's Theorem
(18) (x+y)' = x' • y'
 (19) (x•y)' = x' + y'
 Example


X

= [(A'+C) • (B+D')]'
= (A'+C)' + (B+D')'
...
Three Variables DeMorgan's Theorem
(20) (x+y+z)' = x' • y' • z'
 (21) (xyz)' = x' + y' + z‘




EXAMPLE:
 Apply DeMorg...
Universality of NAND Gates

13
Universality of NOR Gates

14
Alternate Logic Gate
Representations

15
Minterms and Maxterms
x y

z

0 0

0

Minterms
Term Designation
x' y’ z' m0

Maxterms
Term
Designation
x+y+z
M0

0 0

1

x...
Canonical FORMS


There are two types of canonical forms:



the sum of minterms
The product of maxterms

17
Sum of minterms


f1 = x'y'z + xy'z' + xyz = m1 + m4 +m7



f2 = x'yz + xy'z + xyz’ + xyz = m3 + m5+
m6 + m7
x
0
0
0
0
1...
Product of maxterms
The complement of f1 is read by forming a
minterm for each combination that
produces a 0 as:
 f1’=x’y...
Example: Sum of Minterms


Express the Boolean function F = A + B'C
in a sum of minterms.



F=A+B'C
= ABC + ABC' + AB'C...
Example: Product of Maxterms


Express the Boolean function F =xy' + yz in
a product of maxterm form.



F = xy' + yz = ...
STANDARD FORMS


There are two types of standard forms:



the sum of products (SOP)
The product of sums (POS).

22
Sum of Products
The sum of products is a Boolean
expression containing AND terms, called
product terms, of one or more lit...
Product of Sums
A product of sums is a Boolean expression
containing OR terms, called sum terms.
Each term may have any nu...
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Logic Design - Chapter 3: Boolean Algebra

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- Describing Logic Circuits Algebraically.
- rules of evaluating a Boolean expression.
- Boolean Theorems.
- DeMorgan's Theorem.
- Universality of NAND and NOR Gates.
- Alternate Logic Gate Representations.
- Minterms and Maxterms.
- STANDARD FORMS.

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  • We can say that X = A’BC (A’D’).
  • X = [D+ ((A+B)C)']•E = [D+((A+B)’+C’)]•E = [D+(A’B’+C’)]•E = [1+(1•1+0)]•1 = [1+(1+0)]•1 = [1+1] •1 = 1•1 = 1. 
  • X in the first diagram can = A’BC (A’D’)
  • Commutative = Eltabdel - Associative = El2edmag - Distributive = Eltawze3 - Absorption = El2emtsas (mass)
    Proof:- x+xy = x.1+x.y = x(1+y) = x.1 = x 
  • AND will be OR and nwza3 el bar 3lehom el 2, w el3ks sa7e7 
  • Universal because it do the 3 operation INVERTER, AND & OR
  • Universal because it do the 3 operation INVERTER, OR & AND.
  • Logic Design - Chapter 3: Boolean Algebra

    1. 1. CHAPTER 3 Boolean Algebra
    2. 2. Contents         Describing Logic Circuits Algebraically rules of evaluating a Boolean expression Boolean Theorems DeMorgan's Theorem Universality of NAND and NOR Gates Alternate Logic Gate Representations Minterms and Maxterms STANDARD FORMS 2
    3. 3. Describing Logic Circuits Algebraically  OR gate, AND gate, and NOT circuit are the basic building blocks of digital systems 3
    4. 4. Circuits containing Inverters 4
    5. 5. Evaluating Logic Circuit Outputs  Let A=0, B=0, C=1, D=1, E=1 X = [D+ ((A+B)C)'] • E = [1 + ((0+0)1 )'] • 1 = [1 + (0•1)'] • 1 = [1+ 0'] •1 = [1+ 1 ] • 1 =1 5
    6. 6. Rules of evaluating a Boolean expression First, perform all inversions of single terms; that is, 0 = 1 or 1 = 0.  Then perform all operations within parentheses.  Perform an AND operation before an OR operation unless parentheses indicate otherwise.  If an expression has a bar over it, perform the operations of the expression first and then invert the result.  6
    7. 7. Determining Output Level from a Diagram 7
    8. 8. Boolean Theorems 8
    9. 9. Multivariable Theorems (9) (10) (11) (12) (13.a) (13.b) (13.c) (14) (15) (16) x + y = y + x (Commutative law) x • y = y • x (Commutative law) x+ (y+z) = (x+y) +z = x+y+z (Associative law) x (yz) = (xy) z = xyz (Associative law) x (y+z) = xy + xz (Distributive law) x + yz = (x + y) (x + z) (Distributive law) (w+x)(y+z) = wy + xy + wz + xz x + xy = x (Absorption) [proof] x + x'y = x + y (x +y)(x + z) = x +yz 9
    10. 10. Proof of (14, 15, 16) x + xy x + x’y = x (1+y) = x • 1 [using theorem (6)] = x [using theorem (2)] = ( x + x’) (x + y) [theorem 13b] = 1 (x +y) = (x + y) (x +y)(x + z) =xx + xz + yx + yz = x + xz + yx + yz = x (1+z+y) +yz = x . 1 + yz = x + yz 10
    11. 11. DeMorgan's Theorem (18) (x+y)' = x' • y'  (19) (x•y)' = x' + y'  Example  X = [(A'+C) • (B+D')]' = (A'+C)' + (B+D')' = (AC') + (B'D) = AC' + B'D 11
    12. 12. Three Variables DeMorgan's Theorem (20) (x+y+z)' = x' • y' • z'  (21) (xyz)' = x' + y' + z‘   EXAMPLE:  Apply DeMorgan’s theorems to each of the following expressions:  (a) ( A + B + C) D  (b) ABC + DEF  (c) A B + CD + EF 12
    13. 13. Universality of NAND Gates 13
    14. 14. Universality of NOR Gates 14
    15. 15. Alternate Logic Gate Representations 15
    16. 16. Minterms and Maxterms x y z 0 0 0 Minterms Term Designation x' y’ z' m0 Maxterms Term Designation x+y+z M0 0 0 1 x' y' z M1 0 1 0 x' y z’ m2 0 1 1 x' y z m3 1 0 0 x y' z’ m4 1 0 1 x y' z m5 1 1 0 x y z’ m6 1 1 1 xyz m7 x+y+z’ +y x+y’+z +y -t- Z x+y’+z’ + y' +2 x'+y+z +y' + '+y+z’ xt , '+y + '+y’+z x z 4-'+y’+z’ x 2' M1 M2 M3 M4 M5 M6 M7 2 16
    17. 17. Canonical FORMS  There are two types of canonical forms:   the sum of minterms The product of maxterms 17
    18. 18. Sum of minterms  f1 = x'y'z + xy'z' + xyz = m1 + m4 +m7  f2 = x'yz + xy'z + xyz’ + xyz = m3 + m5+ m6 + m7 x 0 0 0 0 1 1 1 1 y 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 f1 0 1 0 0 1 0 0 1 f2 0 0 0 1 0 1 1 1 18
    19. 19. Product of maxterms The complement of f1 is read by forming a minterm for each combination that produces a 0 as:  f1’=x’y’z’ + x’yz’ + x’yz + xy’z + xyz’   f1 = (x + y + z)(x + y' + z)(x + y' + z' )(x’+ y + z)(x’ + y' + z) = Mo M2 M3 M5 M6 Similarly  f2 = ?  19
    20. 20. Example: Sum of Minterms  Express the Boolean function F = A + B'C in a sum of minterms.  F=A+B'C = ABC + ABC' + AB'C + AB'C' + AB'C + A'B'C 20
    21. 21. Example: Product of Maxterms  Express the Boolean function F =xy' + yz in a product of maxterm form.  F = xy' + yz = (xy' + y)(xy' + z) = (x + y)(y' + y)(x + z)(y' + z) = (x + y)(x + z)(y' + z) = (x + y + zz')(x + yy' + z)(xx' + y' + z) = (x + y + z)(x + y + z')(x+y + z)(x+y’+ z)(x + y' + z)(x'+y'+z) = (x + y + z)(x + y + z') (x + y' + z) (x'+y'+z) = M0 M1 M2 M6 = Π (0,1,2,6) 21
    22. 22. STANDARD FORMS  There are two types of standard forms:   the sum of products (SOP) The product of sums (POS). 22
    23. 23. Sum of Products The sum of products is a Boolean expression containing AND terms, called product terms, of one or more literals each. The sum denotes the ORing of these terms.  F = xy + z +xy'z'. (SOP)  23
    24. 24. Product of Sums A product of sums is a Boolean expression containing OR terms, called sum terms. Each term may have any number of literals. The product denotes the ANDing of these terms.  F = z(x+y)(x+y+z) (POS)  F = x (xy' + zy) (nonstandard form)  24
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