# BOOLEAN ALGEBRA: DEFINITION, RULES, THEOREMS, AND EXAMPLES

Boolean algebra is the study of algebra in which the variable’s values are the truth values, and gives output in true and false, normally denoted by 1 and 0 respectively. It is used to understand and solve digital circuits or digital gates. It is also called Binary Algebra.

It has great importance in the development of digital electronics and it is provided in all new programming languages. It is also used in different branches of math like set theory and statistics.

Basic operations performed in __Boolean algebra__ are – conjunction (∧) also named AND operation and has precedence Middle, and disjunction (∨) also named OR, and has the Lowest precedence and negation (¬) about which we knew NOT Operation and has precedence Highest.

Moreover, here we will define Boolean algebra with examples rules important theorems, and methods to solve algebraic expressions.

**Definition of Boolean algebra:**

George Boole invented Boolean algebra in 1854 and it is defined as:

“Boolean algebra is a system of mathematical logic and it represents the relationships between two entities ideas, or objects”

**Rules: **

- Two values are used for variables. To Show the value High Binary 1 and for Low Binary 0.
- The complement of a variable is represented by a bar on it. For example, the complement of B is denoted by .
- OR of the variable is used for addition and it is represented by a plus (+) sign between them. For example, the OR of A, B, and C is represented by A+B+C.
- AND of the Variable is used for multiplication and it is represented by a dot (.) between them. Such as A.B. Normally dot also should be removed like ABC.

**Theorems:**

Using Boolean algebra theorems change the form of a Boolean expression.

There are Five Boolean algebraic Theorems given below with their equations.

### 1. **De Morgan’s Theorem:**

De Morgan’s rule shows the two most important rules of Boolean algebra

### 2. **Transposition Theorem:**

It states that to find the Transposition of algebraic expression you can apply this method

AB + A’C = (A + C) (A’ + B)

### 3. **Redundancy Theorem:**

It states that to find the redundancy of algebraic expression you can apply this method

AB + BC’ + AC = AC + BC’

### 4. **Duality Theorem:**

It states that to find the duality of algebraic expression you can apply this method

Dual of A(B + C) = A + (B.C) = (A + B) (A + C)

### 5. **Complementary Theorem**

It states that to find a complement of algebraic expression you can apply this method

Complement of A(B + C) = A’ + (B’.C’) = (A’ + B’) (A’ + C’)

A __Boolean algebra calculator__ can be used to solve the problems of binary algebra according to the theorems.

**Example Section:**

In this section, we will discuss the theoretical and numerical examples to elaborate Boolean algebra.

**Section1:**

**Example 1:**

The strategy of the counterexample is that if in one situation the statement does not hold, then the statement is false. In practical life, an example is “Albert has never told a lie.”

Now to show this statement is true, you have to provide “proof” that Albert has never told a lie by checking every statement Albert has ever made. So, to prove this statement wrong, you have to focus on all the points you only need to show one lie that Albert has ever spoken.

**Example 2:**

“If subtraction is commutative.” then addition and multiplication should be commutative. That is, excluding imaginary numbers we start from any real number like a, b using a + b= b + a, and a * b = b * a.

However, a counterexample proving this is: 6 – 5 does not equal 5 – 6. Hence, subtraction is not commutative.

**Section 2:**

Truth Tables in Boolean algebra

There are few truth tables through which we easily solve the relevant problems.

**Truth Table-1**

**Construct a truth table for (P****)****).**

P | Q | R | P | Q | (P)) |

T | T | T | T | T | T |

T | T | F | T | F | F |

T | F | T | F | T | F |

F | T | F | T | F | F |

F | F | T | T | T | T |

F | F | F | T | T | T |

There are some numerical examples given below.

**Truth Table-2**

Proof by Perfect induction Method

For the R.H.S. (X + Y) (X+ Z)

And for L.H.S. X + YZ

X | Y | Z | X+Y | X+Z | YZ | (X+Y)(X+Z) | X+YZ |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |

0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |

1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |

1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

**Truth Table-3**

Proof by perfection induction method:

For L.H.S X + Y and for R.H.S. X + Y

X | Y | X+Y | |||

0 | 0 | 1 | 0 | 0 | 0 |

0 | 1 | 1 | 1 | 1 | 1 |

1 | 0 | 0 | 0 | 1 | 1 |

1 | 1 | 0 | 1 | 1 | 1 |

**Remarks: **

(1) A product of sum expression is obtained as follows: each row of the truth table for which the output is 0, the Boolean term is the sum of the variables that are equal to 0 plus the complement of the variables that are equal to 1.

**Summary:**

In this article, the detail of Boolean algebra is studied, and we have tried to cover the maximum important concept of Boolean algebra with the help of its definition, rules, and theorems to solve the expressions. The methods to change forms are also discussed.

Moreover, it is an effective way to solve algebraic expressions after reading and understanding this article, you can easily defend this topic and can find the solutions to the problems related to Boolean algebra.