Boolean Algebra

Boolean algebra is a field in mathematics that focuses on binary variables and deals with only two values, 0 and 1. In 1854, George Boole, an English mathematician, introduced this field of algebra to the mathematical world.

Computer science, artificial intelligence, and engineering are some of the real-world applications that are founded using Boolean algebra. It is offers a mathematical framework for explaining logical operations and expressions.

The three main logical operations in Boolean algebra are conjunction, disjunction, and negation. 

In Boolean algebra, the expressions are the mathematical statements that use logical operators like AND, OR, NOT, XOR, and others. The two possible outcomes for these logical statements are true or false. The values 1 and 0 are used to indicate how inputs and outputs of digital circuits and logic gates are processed. The basic Boolean expressions along with their logical operations are listed below. 

• The Boolean expression for the AND operation (conjunction) is A · B (or A ∧ B). 
   
• The expression for the OR operation is (disjunction) A + B (or A ∨ B)
   
• The expression for the NOT operation (negation) is  ¬A (or A’)
   

By using logical operators like AND, OR, and NOT, we can represent operations in Boolean algebra. The three basic operations are conjunction, disjunction, and negation. Let’s examine each of them in detail. 
 

• Conjunction or AND operation
   

The symbol "•" represents the AND operator in a Boolean expression. It expresses the multiplication of binary numbers. In this operation, if any of the binary variables are false, then the result will be false. When all the variables are true, then the output is also true. The following are the rules of AND operation:

         0 . 0 = 0 or if A = False, B = False, then A . B = False
       
         0 . 1 = 0 or if A = False, B =True, then A . B = False

         1 . 0 = 0 or if A = True, B =False, then A . B = False 

         1 . 1 = 1 or if A = True, B = True, then A . B = True
 

• Disjunction or OR operation 
   

In Boolean algebra, the "+" symbol represents the disjunction or OR operator. It expresses the addition of binary numbers. In this operation, if both of the binary variables are false, the result will be false. The following are the rules of OR operation: 
      
       0 + 0 = 0 or if A = False, B = False, then A + B = False

       0 + 1 = 1 or if A = False, B = True, then A + B = True

      1 + 0 = 1 or if A = True, B = False, then A + B = True

      1 + 1 =1 or if A = True, B = True, then A + B = True
 

• Negation or NOT operation
   

In this operation, if the input is true then it returns false. Likewise, if the input is false, the output is true. An overline represents the variable, (for example,¬A or A'). The following are the rules of NOT operation:
    If A = 1, then (A') = 0. 
    If A = 0, then (A') = 1
 

Boolean algebra is used to design and simplify logic circuits and focuses on logical operations and binary variables. Boolean algebra has some important laws to remember.  
 

• The distributive law:

This law states the distributions of AND over OR, and OR over AND. The distributive law states that when performing the AND operation on two variables and then OR the result with another variable. The result will be equivalent to the third variable’s AND of its OR with each of the first two variables. The Boolean expression of this will be as: 
A + B.C = (A + B) (A + C)
Thus, we can conclude that AND distributes over OR.. 

When performing the OR operation with two variables first and then AND the result with another variable, it is the same as taking the OR of the AND of the third variable with the other two variables. The expression is given as: 
A .(B+C) = (A.B) + (A.C)
Therefore, AND distributes over OR.

• Associative law:
   

This law states that when OR'd or AND'd more than two variables, the way these variables are grouped doesn’t matter in both OR and AND operations. The result will remain unchanged regardless of its grouping order. The expression of the law is:

For OR operation: A + (B + C) = (A + B) + C
For AND operation: A.(B.C) = (A.B).C
 

• Commutative law:

Binary variables of Boolean algebra can only have one of two possible values, either 0 or 1. The commutative law regulates the binary variables. This law states that if we change the position of Boolean variables A and B, it does not change the final output. If we switch the order of the operands from AND to OR or OR to AND, the result of the equation will be the same. The following are the expressions of this law: 

For OR operation: A + B = B + A
For AND operation: A.B = B.A      

• Absorption law:
   

This law simplifies complicated expressions by absorbing the like variables, and the absorption law connects binary variables. The four statements under this law are:

A + A.B = A
A (A + B) = A
A + Ā.B = A + B
A.(Ā + B) = A.B

• Identity law:
   

In both AND (.) and OR(+) operations, we have identity elements. They do not change the result when these variables operate with AND or OR operation. That is expressed as:
A + 0 = A
A.1 = A
 

• Inversion Law:

In Boolean algebra, the inversion law is unique. This law states that, the complement of a complement of a variable results in the variable itself. The mathematical expression of this law is:(A’)’ = A

De Morgan’s theorem is considered one of the most significant theorems in Boolean algebra. Expressions related to the AND, OR, and NOT operators can be simplified with the help of two statements. The two statements are given below: 

The first theorem states that the negation of two Boolean expressions that are AND’d is equal to the OR of the negation of each Boolean variable. The mathematical expression is: 
(A.B)' = A' + B' 
Here, the complement of the product (AND) of two Boolean expressions (A.B)' is equal to the sum (OR) of each negated variable ( A' and B'). 

The second theorem states that the complement of the OR operation between two Boolean variables is equal to the AND operation of their individual complements. The expression is:

(A + B)' = A'. B' NOT A OR B is the same as NOT A AND NOT B.

 

• Boolean algebra is a subfield in mathematics that focuses the logical operations on variables. It has only two possible values, they are either 1 or 0. Boolean algebra plays a crucial role in building digital circuits for computers, robots, and other electronic devices. Logic gates are the decision-makers for any digital circuits. They are the fundamental components that combine various inputs and give outputs based on logical operations and rules of Boolean algebra. For example, A and B are the two inputs and R is the output. Here, some logic gates of Boolean algebra are listed. 
   
• AND gate: The expression of AND gate is expressed as:   
  R = A.B 
  Here, if both of the inputs (A and B) are true, then the output (R) will be true. On the other hand, suppose we have a room light and it has two switches. According to the AND gate, the light will turn on only if we switch on both switches. This is a 2-input AND gate. 
  
   
• OR gate: The Boolean expression of OR gate is: 
   R = A + B 
  Here, if any of the inputs, either A or B is true, then the R will be true. For example, we have a television with two remotes to turn it on. In this case, the TV can be turned on if either of the remotes is used. Here is the image of a 2-input OR gate: 
  
  NOT gate: This gate implies that if we want a true output, then the input should be false. This is known as an inverter. The Boolean equation of NOT gate is: 
   R = Ā
  In this case, the output is the opposite of the input. 
  
   
•  NAND gate: This gate combines the NOT and AND gates. R = (A.B)' is the Boolean equation of the NAND gate. If both the inputs (A and B) are true, then the output R will be false. Here is a 2-input NAND gate:  
  
   
• NOR gate: This gate combines the NOT and OR operations. The Boolean expression for this gate is: 
  R = (A + B)'
  This means if both inputs, A and B are false, then R is true. 
  
   
• EX-OR gate: R = A ⊕ B is the Boolean equation of this gate. By combining AND, NOT, and the OR gate, the EX-OR gate is created. It is known as the exclusive OR gate. If any of the A and B inputs are true, then the output R will be true.
  
   
• EX-NOR gate: The Boolean equation of this gate is R = (A ⊕ B)'. This is the complement of EX-OR gate. If both the A and B inputs are either true or false, then only the R will be true. 

Boolean algebra truth table is a table that shows whether the expression or the output is true or false for the given input variables. Only binary inputs and outputs are included in the truth table. For each logic gate, there is a different truth table.

The truth table of AND gate:


The truth table of OR gate is:


The truth table of NOT gate is:


The truth table of NAND gate is: 


The truth table of the NOR gate is: 


The truth table of the EX-OR gate is: 


The truth table of the EX-NOR gate is: 
 

In the fields of electronic engineering, computer science, artificial engineering, and algebra, the concept of Boolean algebra is very relevant and helpful. The real-life applications of this concept are countless. 
 

• Boolean algebra is used to design digital circuits that are the backbone of electronic devices like computers, mobile phones, and calculators. They use this concept to process the data and make statements and decisions.
  
   
• Tech people use Boolean algebra for coding, finding anything unusual happening on their networks, and checking whether a mail is spam or not. 
  
   
• Boolean algebra can be used to set automation tools to control our homes, machinery, and devices like mobile phones, televisions, and other devices. 
  
   
• Students use this subfield of algebra to solve complex mathematical problems and to filter their results. Venn diagrams use the Boolean algebra to analyze the data.