Boolean Expression in Boolean Algebra

A boolean expression can be seen in boolean algebra. A boolean algebra logically works on discretes values and this logic is invented by “George Boole” hence, came to be known as Boolean algebra. Boolean algebra is a branch of mathematics that deals with logical operations and variables. Boolean algebra, a system of mathematical logic that represents relationships between entities—either ideas or objects. Today, Boolean algebra is of significance to the theory of probability, geometry of sets, and information theory. Furthermore, it is the basis for the design of circuits used in electronic digital computers.

The advantage of Boolean algebra is that it is valid when boolean values i.e, The truth or false of a logical statement are used as variables instead of the numeric quantities given by ordinary algebra. It lends itself to evaluating the logics that are either true (with truth value 1) or false (with truth value 0). The values 0 and 1 are the boolean numbers.

The Boolean variables are represented as binary numbers to represent truths: 1 = true and 0 = false. Boolean conditions which are used in boolean expressions are those which define the statement condition. When given conditional statements are equivalent with truth values, then it can be referred to as equivalent boolean expressions.

Definition:

What is a Boolean Expression?

A boolean expression is an expression that evaluates either true or false. Boolean expressions may be statement, algebra, expression of logic circuit etc.

Boolean Expressions Examples:

1. 8>5.

In the above boolean expression, the value 8 is greater than 5, hence the expression is evaluated as true.

2. 12 is greater than or equal to 8.

In the above boolean expression, the value 12 is greater than 8 but not equal, hence the expression is evaluated as false.

3. 15 is less than 7.

In the above boolean expression, the value 15 is greater than 7, hence the expression is evaluated as false.

Boolean algebra has only two mathematical operations, addition and multiplication. These operations are associated with the OR gate and the AND gate, respectively. 

Logical Addition: 

When the + (the logical addition) symbol is placed between two variables, say A and B, since both A and B can take only the value 0 and 1, we can define the + Symbol by listing all possible combinations for A and B and the resulting value of A + B.

The possible input and output combinations may arranged as follows: 

• 0 + 0 = 0 

• 0 + 1 =1 

• 1 + 0 = 1 

• 1 + 1 = 1 

Logical Multiplication:

We can define the "." (logical multiplication) symbol or AND operator by listing all possible combinations for (input) variables X and Y and the resulting (output) value of A. B as, 

• 0 .0= 0 

• 0 .1 = 0 

• 1 .0 = 0 

• 1 .1 = 1 

Laws of Boolean Algebra:

Six of the basic laws of Boolean algebra are the same as in ordinary algebra,

• Commutative law

• Associative law

• Distributive law

• AND law

• OR law

• Inversion law

The commutative law for addition and multiplication of two variables is given as, 

A + B = B + A And A . B = B . A 

The associative law for addition and multiplication of three variables is written as, 

(A + B) + C = A + (B + C) And (A .B) . C = A. (B. C) 

The distributive law for three variables involves, both addition and multiplication and is written as, A (B+ C) = A B + AC 

AND law is represented by multiplication terminology.  AND laws include 

• A.0 = 0

• A.1 = A

• A.A = A

• A.A` = 0

OR law is represented by addition terminology.  OR laws include 

• A+0 = A

• A+1 = 1

• A+A = A

• A+A` = 1

Inversion  law is represented by true or false values. Double inversion of values gives the same value.

(A`)` = A

Boolean Algebra Theorem:

The two important theorems which are used in Boolean algebra are Demorgan’s First law and De Morgan’s second law. These two theorems are used to change the boolean expression. This theorem basically helps to reduce the given boolean expression in the simplified form. These two Demorgan’s laws are used to change the expression from one form to another form.

1.Demorgan’s First law:

Law states that the complement of the product of the variables is equal to the sum of their individual complements of a variable.

i.e, (A.B)` = A` + B`

Truth table is given by

2.De Morgan's Second law:

Law states that the complement of the sum of the variables is equal to the product of their individual complements of a variable.

i.e, (A+B)` = A`.B`

Truth table is given by

Conclusion: 

We learned an interesting concept of the boolean algebra along with its definition. We saw real-time examples of boolean expressions and values. We learned the laws of boolean algebra and theorems.  We prepared this lesson so that it is easy to understand, but will stick with the students forever as well.