2. DEFINITION
• Boolean Algebra is used to analyze and simplify the digital (logic)
circuits. It uses only the binary numbers i.e. 0 and 1.
• It is also called as Binary Algebra or logical Algebra.
• Boolean algebra was invented by George Boole in 1854.
3. RULE IN BOOLEAN ALGEBRA
Following are the important rules used in Boolean algebra.
• Variable used can have only two values. Binary 1 for HIGH and Binary 0 for
LOW.
• Complement of a variable is represented by an overbar (-). Thus, complement
of variable B is represented as B Bar. Thus if B = 0 then B Bar = 1 and B = 1
then B Bar = 0.
• ORing of the variables is represented by a plus (+) sign between them. For
example ORing of A, B, C is represented as A + B + C.
• Logical ANDing of the two or more variable is represented by writing a dot
between them such as A.B.C. Sometime the dot may be omitted like ABC.
4. BOOLEAN LAWS / THEOREMS
• There are six types of Boolean Laws.
i. Commutative law
• Any binary operation which satisfies the following expression is referred to as
commutative operation.
• Commutative law states that changing the sequence of the variables does not
have any effect on the output of a logic circuit.
A.B = B.A A + B = B + A
5. CONT…
ii. Associative law
• This law states that the order in which the logic operations are performed is
irrelevant as their effect is the same.
ii. Distributive law
• Distributive law states the following condition.
(A.B).C = A.(B.C) (A + B) + C = A + (B + C)
A.(B + C) = A.B +A.C
6. CONT…
iv. AND law
• These laws use the AND operation. Therefore they are called as AND laws.
iv. OR law
• These laws use the OR operation. Therefore they are called as OR laws.
A.1 = 1
A.A’ = 0
A.0 = 0
A.A = A
A + 1 = 1
A + A’ = 0
A + 0 = A
A + A = A
7. CONT…
vi. INVERSION law
• This law uses the NOT operation. The inversion law states that double
inversion of a variable results in the original variable itself.
A = A
8. DE’ MORGAN THEOREMS
• De Morgan has suggested two theorems which are extremely useful
in Boolean Algebra. The two theorems are discussed below.
Theorem 1
The complement of a product of variables is equal to the sum of the complements
of the variables,
Stated another way,
The complement of two or more ANDed variables is equivalent to the OR of the
complements of the individual variables.
A + B = A . B …….. (1)
9. CONT…
Theorem 2
The complement of a sum of variables is equal to the product of the complements
of the variables.
Stated another way,
The complement of two or more ORed variables is equivalent to the AND of the
complements of the individual variables,
A.B = A + B …….. (2)
12. EXERCISE 1:
• Given an expression Y = (A + B)(A + C).
a) Draw a combinational logic gate
b) Build a truth table
c) Simplify the expression using Boolean Theorem
d) Draw a combinational logic gate based on the answer on (c)
e) Build a truth table based on simplify expression.
13. EXERCISE 2:
• Given an expression Y = AB’ + AB + BC
a) Draw a combinational logic gate
b) Build a truth table
c) Simplify the expression using Boolean Theorem
d) Draw a combinational logic gate based on the answer on (c)
e) Build a truth table based on simplify expression.
14. EXERCISE 3:
• Given an expression Y = A’B + BC’ + BC + AB’C’
a) Draw a combinational logic gate
b) Build a truth table
c) Simplify the expression using Boolean Theorem
d) Draw a combinational logic gate based on the answer on (c)
e) Build a truth table based on simplify expression.
15. EXERCISE 4:
• Given an expression Y = A’BC + AB’C + ABC’ + ABC
a) Draw a combinational logic gate
b) Build a truth table
c) Simplify the expression using Boolean Theorem
d) Draw a combinational logic gate based on the answer on (c)
e) Build a truth table based on simplify expression.
17. KARNAUGH MAP (K-MAP)
• Karnaugh Maps offer a graphical method of reducing a digital circuit
to its minimum number of gates.
• The map is a simple table containing 1s and 0s that can express a
truth table or complex Boolean expression describing the operation
of a digital circuit.
• The map is then used to work out the minimum number of gates
needed, by graphical means rather than by algebra.
• Karnaugh maps can be used on small circuits having two or three
inputs as an alternative to Boolean algebra, and on more complex
circuits having up to 6 inputs, it can provide quicker and simpler
minimisation than Boolean algebra.
18. CONT…
Constructing Karnaugh Maps
• The shape and size of the map is dependent on the number of binary
inputs in the circuit to be analysed. The map needs one cell for each
possible binary word applied to the inputs.
19. CONT…
Steps to solve expression using K-map:
i. Select K-map according to the number of variables.
ii. Identify minterms or maxterms as given in problem.
iii. For SOP put 1’s in blocks of K-map respective to the minterms (0’s
elsewhere).
iv. For POS put 0’s in blocks of K-map respective to the maxterms(1’s
elsewhere).
v. Make rectangular groups containing total terms in power of two like
2,4,8 ..(except 1) and try to cover as many elements as you can in one
group.
vi. From the groups made in step 5 find the product terms and sum them
up for SOP form.
25. EXERCISE 9:
INPUT OUTPUT
X Y Z P
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 0
Transfer the truth table below to the Karnaugh Map and write an expression.
25
26. EXERCISE 10:
Using Karnaugh map, proof that an expression:
Y = AB’C’ + A’BC’ + ABC’ + AB’C + A’BC + ABC is Y = A + A’B
26
27. SIMULATOR:
• Boolean Algebra Simplifier (bool-simplifier.com)
• Online Karnaugh map solver with circuit for up to 6 variables
(32x8.com)
• K-Map Generator SOP POS Implicants (aashpointo.github.io)
• Karnaugh Map Solver (charlie-coleman.com)
• 2 Variables (Karnaugh's) K-Map Solver with Steps (getcalc.com)
• karnaugh map karnak truth table Veitch diagram, is a method to
simplify Boolean algebra expressions. The Karnaugh map reduces the
need for extensive calculations example problems (leventozturk.com)
