Boolean Algebra was introduced by George Boole in 1854 and uses values, variables and expressions.
True and False values are represented by one and Zero respectively.

Values are represented by letters and consist of one of the two values either 0 or 1. Operations are functions of one or more variables.

Operations are represented as follows:

AND - X . Y
OR - X + Y
NOT - X'
The above basic operations are combined to give expressions.

Boolean expression is an expression resulting in a boolean value where the value can be either true or false. Boolean expression makes the decision.

For example, (2 < 5) is a Boolean expression as the result is TRUE.

Boolean expressions are also known as comparison expressions, conditional expressions and relational expressions.
An algebraic structure is defined by a set B = {0, 1}, together with two binary operators (+ and .) and a unary operator (-).

Here are the boolean algebra identities:

Identity Name
AND Form
OR Form
Identity law 1 . x = x 0 + x = x
Null law 0 . x = 0
1 + x = 1
Idempotent law x . x = x x + x = x
Inverse law x . $\bar{x}$ = 0 x + $\bar{x}$ = 1
Commutative law x . y = yx x + y = y + x
Associative law (x . y) . z = x(yz) (x + y) + z = x + (y + z)
Distributive law x + yz = (x + y)(x + z) x(y + z) = xy + yz
Absorption law x . (x + y) = x x(x + y) = x
Demorgan's law
(x . y)' = x' + y'
(x + y)' = x' . y'

Basic logical operators are the logic functions AND, OR, NOT and they operate on binary values and binary variables. Logic gates implement logic functions.

The three basic logical operations are:

AND- Denoted by a dot (.)
For example, Y = A . B is read as Y is equal to A AND B.

OR- Denoted by a plus (+)
For example, Z = X + Y is read as Z is equal to X OR Y.

NOT- Denoted by a single quote mark ( ' ) or ($\sim$) before the variable.
For example, X = A' is read as X is equal to NOT A.

Digital systems are constructed using logic gates. The basic gates are described below:

AND gate
The AND gate is an electronic circuit that gives a high output (1) only if all its inputs are high. Sometimes dot is omitted and it is written as AB instead of A . B.

And Gate

OR gate
The OR gate is an electronic circuit that gives a high output (1) if one or more of its inputs are high.


Or Gate

NOT gate
The NOT gate is an electronic circuit which gives an inverted version of the input at its output. It is also known as an Inverter.

Not Gate
Operations are defined on the values '0' and '1' for each operator. A truth table is a mathematical table used in logic with boolean algebra, boolean functions and propositional calculus.

AND truth table
X
Y
Z = X . Y
0
0 0
0 1 0
1
0 0
1 1 1

Mathematical operation is denoted by A $\cup$ B.

OR truth table
X
Y
Z = X + Y
0 0 0
0 1 1
1 0 1
1 1 1

If atleast one input is 1 then the output observed is 1. Mathematical operation is denoted by A $\cap$ B.

NOT truth table
X Z = X'
0 1
1 0


Solved Examples

Question 1: Find the Boolean algebra expression for the following system.

Boolean Algebra Example
Solution:
 
The system consists of an AND Gate, a NOR Gate and an OR Gate. The expressions for AND and NOR gates are A . B and A + B.  Both these expressions gives input to the OR gate which is defined as A + B. The final output expression is shown below:

Boolean Algebra Examples

The output of the system is Q = (A . B) + (A + B).
The notation (A + B) is same as De Morgan's notation A . B.
Then substituting A . B gives us Q = (A . B) + (A . B).
Given below is the truth table to represent this.

 Input  Input  Intermediates  Intermediates
 Output
  A    B  
      A . B  
     (A . B)     Q  
  0    0          0          1     1 
  0    1
         0          0     0
  1    0          0          0     0
  1    1          1
         0     1


 

Question 2: Construct a truth table for A + BC
Solution:
 

    A          B    
   C   
   BC  
    A + BC  
    0      0    0     0        0
    0      0    1     0        0
    0      1    0     0        0
    0      1    1     1       1
    1      0    0     0       1
    1      0    1     0       1
    1      1    0     0       1 
    1      1    1     1       1