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 the 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  0  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 