Sum of Products (SOP)
&
Product of SUMS(POS)
SOP & POS
A LOGIC CIRCUIT
A logic circuit contains one or more basic logic gates.
Logic circuits have the properties of Boolean algebra.
An AND-OR logic circuit - Called “ Sum of Products”.
An OR-AND logic circuit -Called “Product of Sums”.
SOP & POS
SIMPLIFICATION OF BOOLEAN
EXPRESSIONS
SOP & POS
BOOLEAN ALGEBRA LAWS AND RULES
Commutative Law
1. Commutative law of addition: A+B = B+A
2. Commutative law of Multiplication
A*B=B*A
SOP & POS
Associative Law:
1. Associative Law of Addition:
A+(B+C)=(A+B)+C
2. Associative Law of Multiplication:
A(BC)=(AB)C
These laws mean that the grouping of several variables Ored or
ANDed together does not matter.
SOP & POS
Distributive Law:
A(B+C)=AB+AC
(A+B) . (C+D)= AC+AD+BC+BD (FOIL method)
Boolean expression simplification
SIMPLIFY
Simplify the Boolean function F=AB+ BC + B′C.
Solution. F = AB + BC + B′C
= AB + C(B + B′)
= AB + C
Boolean expression simplification
SIMPLIFY
Simplify the Boolean function F= A′B′C + A′BC + AB′.
Solution. F = A′B′C + A′BC + AB′
= A′C (B′+B) + AB′
= A′C + AB′
SOP & POS
SIMPLIFY
F= ABC’ + AB’C + A’BC + ABC
[Since X+X = X, ABC can be repeated many times]
= ABC’ + ABC + AB’C + ABC + A’BC+ABC
= AB(C’+C) + AC(B’+B) + BC(A’+A)
= AB+ AC+BC
Simplify : A’B’C+ A’BC +AB’
Karnaugh Maps
(K-Maps)
SOP & POS
Karnaugh map is a method used to simplify Boolean expressions.
The Boolean expressions must be in SOP format before they can be
put into a map.
We can transfer logic values from a Boolean statement or a truth
table into a Karnaugh map
SOP & POS
It contains boxes called cells
Each cell represents one of the 2n possible products that can be
formed from n variables
Two variable map contains 4 cells (22), 3 variable map contains 8
cells(23), 4 variable map contains 16 cells(24)
2 Variable cells
A B
0
1
0
00
A’B’ 0
01
A’B
1
1
10
AB’ 2
11
AB
3
SOP & POS
3 Variable cells
C AB
00
11
10
010
110
A’B’C’ 0
A’BC’ 2
ABC’ 6
100
4
AB’C’
001
011
111
101
A’B’C 1
A’BC 3
ABC 7
AB’C 5
000
0
01
1
SOP & POS
The Karnaugh map is completed by entering a '1‘(or ‘0’) in each of
the appropriate cells
Within the map, adjacent cells containing 1's (or 0’s) are grouped
together in twos, fours, or eights
SOP & POS
SOP & POS
SOP & POS
SOP & POS
SOP & POS
Simplify the Boolean function
F = A′BC + A′BC′ + AB′C′ + AB′C.
A
BC
00
01
11
10
0
1
1
1
= A’B + AB’
1
1
SOP & POS
Simplify the Boolean function
F (A, B, C) = Σ (0, 2, 4, 5, 6).
A
BC
00
01
11
10
0
1
1
1
1
=AB’+C’
1
1