- 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.

1. CHAPTER 3
Boolean Algebra

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. Describing Logic Circuits Algebraically

OR gate, AND gate, and NOT circuit are
the basic building blocks of digital systems
3

4. Circuits containing Inverters
4

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. 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. Determining Output Level from a Diagram
7

8. Boolean
Theorems
8

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. 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. 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. 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. Universality of NAND Gates
13

14. Universality of NOR Gates
14

15. Alternate Logic Gate
Representations
15

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. Canonical FORMS

There are two types of canonical forms:


the sum of minterms
The product of maxterms
17

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. 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. 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. 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. STANDARD FORMS

There are two types of standard forms:


the sum of products (SOP)
The product of sums (POS).
22

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. 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