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BOOLEAN ALGEBRA
Lecture 5
Digital Design
Dr. PO Kimtho
Department of Computer Sciences
Norton University (NU)
Boolean Algebra
Contents
Boolean Operations & Expression
Laws & Rules of Boolean algebra
DeMorgan’s Theorems
Boolean analysis of logic circuits
Simplification using Boolean Algebra
Standard forms of Boolean Expressions
Boolean Expressions & truth tables
Boolean Algebra (Cont.)
The Karnaugh Map
Karnaugh Map SOP minimization
Karnaugh Map POS minimization
Programmable Logic
Boolean Operations & Expression
Expression
Variable
a symbol used to represent logical quantities (1 or 0)
E.g : A, B,..used as variable
Complement
inverse of variable and is indicated by bar over variable
E.g : Ā
Boolean Operations & Expression
Operation
Boolean Addition
– equivalent to the OR operation
X = A + B
Boolean Multiplication
– equivalent to the AND operation
X = A∙B
A
B X
A
B X
Laws of Boolean Algebra
Commutative
Addition & multiplication
Associative
Addition & multiplication
Distributive
Same as ordinary algebra
Commutative law of addition:
A+B = B+A
the order of OR-ing does not matter.
Addition
Commutative Law
Commutative Law
Commutative law of Multiplication
AB = BA
the order of ANDing does not matter.
Multiplication
Associative Law
Associative law of addition
A + (B + C) = (A + B) + C
The grouping of ORed variables does not matter
Addition
Associative Law
Associative law of multiplication
A(BC) = (AB)C
The grouping of ANDed variables does not matter
Multiplication
Distributive Law
A(B + C) = AB + AC
Question: (A+B)(C+D) ?
Rules of Boolean Algebra
Rule 1: A + 0 = A
In math if you add 0 you have changed nothing.
In Boolean Algebra ORing with 0 changes nothing.
Rule 2: A + 1 = 1
ORing with 1 must give a 1 since if any input
is 1 an OR gate will give a 1.
Rules of Boolean Algebra
Rule 3: A . 0 = 0
In math if 0 is multiplied with anything you
get 0. If you AND anything with 0 you get 0.
Rules of Boolean Algebra
Rule 4: A . 1 = A
ANDing anything with 1 will yield the anything.
Rules of Boolean Algebra
Rule 5: A + A = A
ORing with itself will give the same result
Rules of Boolean Algebra
Rule 6: A + A = 1
Either A or A must be 1 so A + A =1
Rules of Boolean Algebra
Rule 7: A . A = A
ANDing with itself will give the same result
Rules of Boolean Algebra
Rule 8: A . A = 0
In digital Logic 1 =0 and 0 =1, so AA=0 since one of the inputs must be 0.
Rules of Boolean Algebra
Rule 9: A = A
If you NOT something twice, you are back to
the beginning
Rules of Boolean Algebra
Rule 10: A + AB = A
Proof:
A + AB = A (1 + B) DISTRIBUTIVE LAW
= A∙1 RULE 2: (1+B) = 1
= A RULE 4: A∙1 = A
Rules of Boolean Algebra
Rule 11: A + AB = A + B
If A is 1 the output is 1 , If A is 0 the output is B
Proof:
A + AB = (A + AB) + AB RULE 10
= (AA + AB) + AB RULE 7
= AA + AB + AA +AB RULE 8
= (A + A)(A + B) FACTORING
= 1∙(A + B) RULE 6
= A + B RULE 4
Rules of Boolean Algebra
Rule 12: (A + B) (A + C)= A + BC
PROOF
(A + B)(A +C) = AA + AC +AB +BC DISTRIBUTIVE LAW
= A + AC + AB + BC RULE 7
= A(1 + C) +AB + BC FACTORING
= A.1 + AB + BC RULE 2
= A(1 + B) + BC FACTORING
= A.1 + BC RULE 2
= A + BC RULE 4
Rules of Boolean Algebra
SUMMARY - LAWS OF BOOLEAN ALGEBRA
[1] COMMUTATIVE:
A + B = B + A
AB = BA
[2] ASSOCIATIVE:
A + (B + C) = (A + B) + C
A(BC) = (AB)C
[3] DISTRIBUTIVE:
A(B + C) = AB + AC
(A + B)(C + D) = AC + AD + BC + BD
Rules of Boolean Algebra
SUMMARY - RULES OF BOOLEAN ALGEBRA
[1] A + 0 = A [7] A.A = A
[2] A + 1 = 1 [8] A.A = 0
[3] A.0 = 0 [9] A = A
[4] A.1 = A [10] A + AB = A
[5] A + A = A [11] A + AB = A + B
[6] A + A = 1 [12] (A + B)(A + C) = A + BC
Rules of Boolean Algebra
[1] A + 0 = A [7] A.A = A
[2] A + 1 = 1 [8] A.A = 0
[3] A.0 = 0 [9] A = A
[4] A.1 = A [10] A + AB = A
[5] A + A = A [11] A + AB = A + B
[6] A + A = 1 [12] (A + B)(A + C) = A + BC
Rules of Boolean Algebra
SUMMARY - RULES OF BOOLEAN ALGEBRA
DeMorgan’s Theorems
Two most important theorems of Boolean Algebra were
contributed by De Morgan
Extremely useful in simplifying expression in which
product or sum (POS) of variables is inverted
The TWO theorems are:
X.Y = X + Y &
X+Y = X . Y
X.Y = X + Y
(a) & (b) Equivalent circuit implied by the
theorem
(c) Alternative symbol for the NAND function
(d) Truth table that illustrates DeMorgan’s
Theorem
(d)
Input Output
X Y XY X+Y
0 0 1 1
0 1 1 1
1 0 1 1
1 1 0 0
DeMorgan’s Theorems
(a) & (b) Equivalent circuit implied by the
theorem
(c) Alternative symbol for the NOR function
(d) Truth table that illustrates DeMorgan’s
Theorem
Input Output
X Y X+Y XY
0 0 1 1
0 1 0 0
1 0 0 0
1 1 0 0 (d)
X+Y = X . Y
DeMorgan’s Theorems
Solve this
Solve this
Standard Forms of Boolean Expression
Sum of products (SOP)
Sum of products (SOP)
Sum of products (SOP)
Product of sum (POS)
Standard Forms of Boolean Expressions
Standard Forms of Boolean Expressions
Solve this
Example (Standard SOP)
Example (Standard POS)
Boolean Expressions & Truth Tables
Boolean Expressions & Truth Tables
Boolean Expressions & Truth Tables
Solve this
The Karnaugh Map (K-Map)
The K-Map
K-Map SOP Minimization
K-map SOP Minimization
K-map SOP Minimization
K-map SOP Minimization
K-map SOP Minimization
K-map SOP Minimization
K-map SOP Minimization
K-map SOP Minimization
K-map SOP Minimization
K-map SOP Minimization
K-map SOP Minimization
K-map SOP Minimization
K-map SOP Minimization
K-map SOP Minimization
K-map SOP Minimization
Example
Example
Example
K-map POS Minimization
K-map POS Minimization
K-map POS Minimization
K-map POS Minimization
K-map Minimization – Don’t cares
K-map Minimization – Don’t cares
Example
Example
Solve this
