Boolean Algebra (Operation, Laws, Calculation, Truth Table)
What is Boolean Algebra?
What are the operations of Boolean algebra?
Boolean Operators 
Notation 
Representation 
Definition 
Conjunction 
^ 
AND 
In this operation, the values are true
when both the terms are true otherwise false. It acts as a product of
numbers. A.B or A^B 
Disjunction 
v 
OR 
In this operation, the values are
false when both the terms are false or otherwise true. It acts as a product
of numbers. A+B or AvB 
Negation 
¬ 
NOT 
It reverts the binary variables such
as if true it transposes it to false and if false it converts it to true. 
What is a truth table in Boolean Algebra?
A truth table for conjunction
A 
B 
A.B or A^B 
1 
1 
1 
1 
0 
0 
0 
1 
0 
0 
0 
0 
A truth table for disjunction
A 
B 
A+B or AvB 
1 
1 
1 
1 
0 
1 
0 
1 
1 
0 
0 
0 
A truth table for negation
A 
¬A 
1 
0 
0 
1 
Theorems of Boolean algebra
Theorems 
Statement 
Expression 
De Morgan’s 1^{st} law 
The 1^{st}
law of De Morgan states that the complement of the product of the variables
is equal to the sum of their individual complements of a variable. 
(X.Y)’ = X’+Y’ 
De Morgan’s 2^{nd} law 
The 2^{nd}
law of De Morgan states that the complement of the sum of variables is equal
to the product of their individual complements of a variable. 
(X+Y)’ = X’.Y’ 
The truth table of De Morgan’s 1st Law
X 
Y 
X’ 
Y’ 
X.Y 
(X.Y)’ 
X’ + Y’ 
0 
0 
1 
1 
0 
1 
1 
0 
1 
1 
0 
0 
1 
1 
1 
0 
0 
1 
0 
1 
1 
1 
1 
0 
0 
1 
0 
0 
The truth table of De Morgan’s 2nd Law
X 
Y 
X’ 
Y’ 
X + Y 
(X + Y)’ 
X’ * Y’ 
0 
0 
1 
1 
0 
1 
1 
0 
1 
1 
0 
1 
0 
0 
1 
0 
0 
1 
1 
0 
0 
1 
1 
0 
0 
1 
0 
0 
Laws of Boolean Algebra
Laws Name 
Definition 
Expression 
Commutative Law 
As a result of commutative law, a
logic circuit's output does not change if its variables are changed in
sequence. 
X * Y = Y * X X + Y = Y + X

Associative Law 
According to this
law, the order in which logic operations are performed has no effect on their
effects. 
(X * Y) * Z = X *(Y*Z) (X + Y) + Z = X+(Y+Z) 
Distributive Law 
This law is used for both addition and
multiplication and states that 
X* (Y + Z) = (X* Y) + (X* Z) X+ (Y * Z) = (X+ Y) * (X+ Z) 
AND Law 
The law that uses AND operation is said
to be the AND law of binary algebra. 
X * 0 = 0 X * 1 = X X * X = X X * x̄=0 
OR Law 
The law that uses the OR operation is
said to be the OR law of binary algebra. 
X + 0 = X X + 1 = 1 X + X = X X +x̄= 1 
Inversion Law 
The inversion law of Boolean algebra states
that double inversion of the original variable produces the original variable. 
x̄̄ = x 
How to calculate Boolean algebra problems?
Example 1
X 
Y 
Z 
X + Y 
X * Z 
X + Z 
(X + Y) + (X * Z) 
(X + Y) + (X * Z) * (X + Z) 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
1 
0 
0 
0 
1 
0 
1 
0 
0 
1 
0 
0 
1 
1 
1 
0 
1 
1 
1 
1 
0 
0 
1 
0 
1 
1 
1 
1 
0 
1 
1 
1 
1 
1 
1 
1 
1 
0 
1 
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1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
Example 2
X 
Y 
Z 
X + Y 
Y + Z 
(X + Y) + Z 
X * (Y + Z) 
[(X + Y) + Z] + [X * (Y + Z)] 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
1 
1 
0 
1 
0 
1 
0 
1 
1 
1 
0 
1 
0 
1 
1 
1 
1 
1 
0 
1 
1 
0 
0 
1 
0 
1 
0 
1 
1 
0 
1 
1 
1 
1 
1 
1 
1 
1 
0 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 