**Boolean Algebra**invention started by George Boole. He invented a

**rule**that there is a clear connection between mathematics and logic.

Previously I have discussed De Morgan's Law.

## Boolean Algebra rules

**Boolean Algebra rules**have been made by depending on true false. But when the use of binary number system(0,1) started on computers from that time this true-false is represented by binary numbers 1 and 0.

Only for addition and multiplication binary digit 0 and 1 are used in Boolean Algebra. No geometrical, trigonometric rule is not usable in

**Boolean algebra**. Here no negative or fractional number is not usable.

For different simplifications, we need to apply many

**rules**. By applying these rules we can make the equations simple.

#### Variable

The value of the Boolean variable is changeable. It changes for time. If A is a Boolean variable then it can take 0 or 1 any of these.

#### Constant

In Boolean algebra, 2 digits (0 and 1) are used. Value of 0 and 1 never changes. That is why 0 and 1 are called Boolean constant is a Boolean algebra.

#### Substitute

Two value of boolean algebra is 0 and 1. These two are the substitute for each other. It is represented by the ' ¯ ' sign. Sometimes ' / ' sign is also used.

#### Substitution rules

- ͞0 =1
- 1 = 0
- Ā̄ = A
- If A = 0 then Ā = 1
- If A = 1 then Ā=0

### Basic operations

3 types of Boolean basic operations:

**Logical addition**

It is also called OR operation. "+" sign is usually used in this case. Example: A OR B is represented as A+B.

**Logical multiplication**

It is also called AND operation. " • " sign is usually used in this case. Example: A AND B is represented as A•B.

**Logical inversion**

It is complementation or NOT operation. " / " or " ˊ " sign is used in this case. Example: NOT (A) is represented as Aˊ.

### Logical operator

In

**Boolean algebra**for completing 3 Boolean operations, we use some signs. Those are called logical operators.AND operator: The logical multiplication process is done by this.

OR operator: The logical addition process is done by this.

NOT operator: Logical inversion is done by this.

### Postulates

All the operations in Boolean algebra are completed by addition and multiplication. In the addition and multiplication process, Boolean algebra follows some rules.

Addition rules

- 0+0 = 0
- 0+1 = 1
- 1+0 = 1
- 1+1 = 1

Multiplication rules

- 0.0 = 0
- 0.1 = 0
- 1.0 = 0
- 1.1 = 1

### Fundamental

(A) A.0 =0

A | 0 | A.0 |

0 | 0 | 0 |

1 | 0 | 0 |

(B) A.1 = A

A | 1 | A.1 |

0 | 1 | 0 |

1 | 1 | 1 |

(C) A.A = A

A | A | A.A |

0 | 0 | 0 |

1 | 1 | 1 |

(D) A.Ā = 0

A | Ā | A.Ā |

0 | 1 | 0 |

1 | 0 | 0 |

(E) A + 0 = A

A | 0 | A+0 |

0 | 0 | 0 |

1 | 0 | 1 |

(F) A + 1 = 1

A | 1 | A+1 |

0 | 1 | 1 |

1 | 1 | 1 |

(G) A+A = A

A | A | A+A |

0 | 0 | 0 |

1 | 1 | 1 |

(G) A+Ā = 1

A | Ā | A+Ā |

0 | 1 | 1 |

1 | 0 | 1 |

** Communicative**

#### A) A+B=B+A

A | B | A+B | B+A |

0 | 0 | 0 | 0 |

0 | 1 | 1 | 1 |

1 | 0 | 1 | 1 |

1 | 1 | 1 | 1 |

#### B) A.B = B.A

A | B | A.B | B.A |

0 | 0 | 0 | 0 |

0 | 1 | 0 | 0 |

1 | 0 | 0 | 0 |

1 | 1 | 1 | 1 |

### Associative

#### (A) A+(B+C) = (A+B) + C

A | B | C | B+C | A+B | A + (B+C) | (A+B)+C |

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

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

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

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

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

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

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

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

#### (B) A(BC) = (AB)C

A B C BC AB A (BC) (AB) C 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 0 0

### Distributive

A | B | C | BC | AB | A (BC) | (AB) C |

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

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

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

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

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

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

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

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

(A) A(B+C) = AB + AC

A | B | C | AB | AC | B+C | A (B+C) | AB+AC |

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

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

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

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

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

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

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

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

#### (B) A+BC = (A+B)(A+C)

A | B | C | BC | A+B | A+C | A+BC | (A+B)(A+C) |

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

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

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

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

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

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

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

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

### Some extra laws

(A) A+AB = AA | B | AB | A + AB |

0 | 0 | 0 | 0 |

0 | 1 | 0 | 0 |

1 | 0 | 0 | 1 |

1 | 1 | 1 | 1 |

(B) Ā̄ = A

A | Ā | Ā̄ |

0 | 1 | 0 |

1 | 0 | 1 |

(C) A(A+B) = A

A | B | A+B | A (A+B) |

0 | 0 | 0 | 0 |

0 | 1 | 1 | 0 |

1 | 0 | 1 | 1 |

1 | 1 | 1 | 1 |

(D) A + ĀB = A+B

A | B | Ā | ĀB | A+ĀB | A+B |

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

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

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

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

(E) A(Ā+B) = AB

A | B | Ā | Ā+B | A(Ā+B) | AB |

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

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

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

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

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