The document discusses digital electronics and Boolean logic. It covers Boolean postulates and laws including commutative, associative, distributive, identity, redundancy, and duality laws. It also discusses Boolean expressions and how to minimize them using Boolean laws and postulates. Additionally, it defines canonical and standard forms for Boolean expressions, describes minterms and maxterms for representing Boolean functions with multiple variables, and provides examples of converting between sum-of-products and product-of-sums forms.

1. Digital Electronics
MINIMIZATION TECHNIQUES AND LOGIC
GATES

2. BOOLEAN POSTULATES AND LAWS:
T1 : Commutative Law
(a) A + B = B + A
(b) A B = B A
T2 : Associate Law
(a) (A + B) + C = A + (B + C)
(b) (A B) C = A (B C)
T3 : Distributive Law
(a) A (B + C) = A B + A C
(b) A + (B C) = (A + B) (A + C)
T4 : Identity Law
(a) A + A = A
(b) A A = A

3. Cont.
T6 : Redundance Law
(a) A + A B = A
(b) A (A + B) = A
T7 :
(a) 0 + A = A
(b) 0 A = 0
T8 :
(a) 1 + A = 1
(b) 1 A = A

4. Cont.

5. Boolean postulates
Boolean postulates are
•The Commutative Law of addition for two variable.
A + B = B + A
•The Commutative Law of multiplication for two variable.
A . B = B . A
•The Associative law of addition with multiplication is written as
A + (B + C) = A +B +C
•The Associative law of multiplication with addition is written as
A . (B . C) = (A . B) . C
•The Associative law of multiplication with addition is written as
A . (B + C) = A . B + A . C
•The Associative law of addition with multiplication is written as
A + (B . C) = (A + B) . (A + C)

6. De-morgan’s Theorem
DE-MORGAN’S THEOREM
1.(AB)' = A' + B'
The complement of a product is equal to the sum of the
complements
2.( A+B)' = A' . B'
The complement of a sum term is equal to the product of
the complements

7. Duality Principle
PRINCIPLE OF DUALITY:
Principle of Duality theorem says,
•Changing each OR operator to an AND operator
•Changing each AND operator to an OR operator
•Replace 0’s by 1’s and 1’s by 0’s
BOOLEAN EXPRESSION:
Boolean expressions are minimized by using Boolean laws
and postulates.

8. Minimization of Boolean
Expressions
MINIMIZATION OF BOOLEAN EXPRESSIONS
Simplify the Boolean expression F=x′y′z′+x′yz+xy′z′+xyz′
To a minimum number of literals
F=x′y′z′+x′yz+xy′z′+xyz′
=x′y′z′+x′yz+xz′(y′+y)
=x′y′z′+x′yz+xz′
=x′yz+ z′(x′y′+x)
= x′yz+z′(x′+x)(y′+x)
F=x′yz+xz′+z′y′

9. Canonical and standard forms
•Sum-of-Product (SOP) form: When two or more
product terms are summed by Boolean addition, the
result is a Sum-of-Product or SOP expression
• Product-of-Sum (POS) form: When two or more sum
terms are multiplied by Boolean multiplication, the
result is a Product-of-Sum or POS expression

10. Cont.
CANONICAL FORM:
In Sum Of Products(SOP) and Product Of Sums(POS), if all the term
contains all the variables either in true or in complementary form then its
said to be canonical SOP or canonical POS.
MINTERM ( canonical SOP)
•In a Boolean function, a binary variable (x) may appear either in its
normal form (x) or in its complement form (x’).
• Consider 2 binary variables x and y and an AND operation, there are 4
and only 4 possible combinations: x’•y’, x’•y, x•y’ & x•y.
•Each of the 4 product terms is called a MINTERM or STANDARD
PRODUCT
•By definition, a Minterm is a product which consists of all the variables
in the normal form or the complement form but NOT BOTH.
e.g. for a function with 2 variables x and y: x•y’ is a minterm but x’ is
NOT a minterm
e.g. for a function with 3 variables x, y and z: x’yz’ is a minterm but
xy’ is NOT a minterm

11. Cont.
MAXTERM (canonical POS)
•Consider 2 binary variables x and y and an OR
operation, there are 4 and only 4 possible combinations:
x’+y’, x’+y, x+y’, x+y.
•Each of the 4 sum terms is called a MAXTERM or
STANDARD SUM.
•By definition, a Maxterm is a sum in which each variable
appears once and only once either in its normal form or
its complement form but NOT BOTH.

12. Minterms and Maxterms for 3
Variables

13. Minterm Boolean Expression
Boolean functions can be expressed with minterms,
e.g.f1(x,y,z) = m1 + m4 + m6 = Σm(1, 4, 6)
f2(x,y,z) = m2 + m4 + m6+ m7 = Σm(2, 4, 6, 7)

14. Maxterm Boolean Expression
Maxterm Boolean Expression
Boolean functions can also be expressed with maxterms,
e.g.f1’ = x’y’z’+x’yz’+x’yz+xy’z+xyz
f1 = (x’y’z’+x’yz’+x’yz+xy’z+xyz)’
= (x+y+z)(x+y’+z)(x+y’+z’)(x’+y+z’)(x’+y’+z’)
= M0•M2•M3•M5•M7
= Π M(0, 2, 3, 5, 7)
f2 = M0•M1•M3•M5 = Π M(0, 1, 3, 5)

15. Truth table for f1 and f2

16. Express Boolean Functions in
Minterms
Express Boolean Functions in Minterms
•If product terms in a Boolean function are not minterms, they can
be converted to minterms
e.g. f(a,b,c) = a’ + bc’ + ab’c
•Function f has 3 variables, therefore, each minterm must have 3
literals.
•Neither a’ nor bc’ are minterms.They can be converted to
minterm.ab’c is a minterm

17. Cont.
Conversion to Minterms
Let the function be f(a,b,c) = a’ + bc’ + ab’c
•To convert a’ to a minterm, the 2 variables (b, c) must
be added, without changing its functionality.
Since a’=a’•1 & 1 = b+b’, a’= a’(b + b’) = a’b + a’b’
•Similarly, a’b = a’b(c + c’) = a’bc + a’bc’ and a’b’ =
a’b’(c+c’) = a’b’c + a’b’c’
•bc’ = bc’(a+a’) = abc’ + a’bc’
•f = a’bc+a’bc’+a’b’c+a’b’c’+abc’+a’bc’+ab’c

18. Express Boolean Functions in
Maxterms
Express Boolean Functions in Maxterms
By using the Distribution Law: x+yz = (x+y)(x+z),
•Boolean function can be converted to an expression
in product of maxterms
e.g. f(a,b,c) = a’+bc’
= (a’+b)(a’+c’) {not maxterms}
= (a’+b+cc’)(a’+c’+bb’) {cc’=0}
= (a’+b+c)(a’+b+c’)(a’+c’+b)(a’+c’+b’)
= (a’+b+c)(a’+b+c’)(a’+c’+b’)

19. Cont.
Express the Boolean function F=A+B′C in sum of
min terms.
Given
F=A+B′C
=A(B+ B′)(C+C′)+ B′C(A+A′)
=(AB+A B′)(C+C′)+B′C(A+A′) =ABC+ABC′+AB′C+AB
′C′+AB′C+A′B′C
= ABC+ABC′+AB′C+AB′C+A′B′C
F=m1+m4+m5+m6+m7

20. Cont.
POS form:
Given
D=(A′+B)(B′+C)
=A′B′+A′C+BB′+BC
= A′B′+A′C+BC
= A′+B′+ A′C+BC
= A′(1+C)+B′+BC
= A′+B′+BC
D= A′B′+BC
Using missed terms formulae;
= A′B′(C+C′)+(A+A′)BC
= A′B′C+ A′B′C′+ABC+A′BC
D(A,B,C)= Σ m(1,0,7,3)

21. Thank You