# Boolean Algebra in digital electronics

## Introduction

Boolean algebra is an algebraic system that expresses the digital circuits. It has two elements 0 and 1, two binary operators AND and OR and one unary operator NOT. We can express any complex logic statement by Boolean algebra. It has some rules and laws to write expressions.

Boolean algebra is different from both binary number system and ordinary algebra. In Boolean algebra A.A =A and A+A=A, while in ordinary algebra $A.A=A^2$ and A+A=2A. In Boolean algebra 1+1=1, while in binary number system 1+1=10 and in ordinary algebra 1+1=2. In Boolean algebra there is no negative and fractional numbers and no subtraction and division. It has only logical operators AND, OR, NOT, NAND, NOR, X-OR, X-NOR.

### Logic Operations

#### AND Operation:

AND operation in Boolean algebra is the ordinary multiplication. It is the logical multiplication as performed by AND gate. The operator for AND operation is (.).

#### OR Operation:

OR operation in Boolean algebra is the ordinary addition. It is the logical addition as performed by OR gate. The operator for OR operation is (+).

#### NOT Operation:

NOT operation in Boolean algebra is complement of the operand. It is same as the operation performed by NOT gate. The operator for NOT operation is bar (_).

#### NAND Operation:

NAND operation is the complement of AND operation and same as the operation of NAND gate.

#### NOR Operation:

NOR operation is the complement of OR gate and same as the operation of NOR gate.

#### X-OR Operation:

X-OR operation using two variables A and B in Boolean algebra can be-

$A\oplus&space;B=\overline{A}B+A\overline{B}$

#### X-NOR Operation:

X-NOR operation using two variables A and B in Boolean algebra can be-

$A\odotB=AB+\overline{A}.\overline{B}$

### Principles of Boolean algebra:

Principles of Boolean algebra are a set of logical expressions that we accept without any proof.

AND operation

Principle 1: 0.0=0

Principle 2: 0.1=0

Principle 3: 1.0=0

Principle 4: 1.1=1

OR operation

Principle 5: 0+0=0

Principle 6: 0+1=1

Principle 7: 1+0=1

Principle 8: 1+1=1

NOT Operation

Principle 9: $\overline{0}=1$

Principle 10: $\overline{1}=0$

### Laws of Boolean algebra:

##### Complementation laws:

Complementation means inversion, i.e. changing 0 to 1 and 1 to 0. Complementation laws are:

Law 1: $\overline{0}=1$

Law 2: $\overline{1}=0$

Law 3: if A=0 then $\overline{A}=1$

Law 4: if A= 1 then $\overline{A}=0$

Law 5: $\overline{\overline{A}}=A$

##### AND laws:

The AND laws are as follows-

Law 1: A.0=0

Law 2: A.1=A

Law 3: A.A=A

Law 4: $A.\overline{A}=0$

##### OR Laws:

THE OR laws are –

Law 1: A+0=A

Law 2: A+1=1

Law 3: A+A=A

Law 4: $A+\overline{A}=1$

##### Commutative laws:

Commutative laws allow to change the position of variables of AND and OR operations.

Law 1: A+B=B+A

LAW 2: A.B=B.A

##### Associative laws:

The associative laws allow making a group of variables.

Law 1: (A+B) +C=A+ (B+C)

LAW 2: (A.B) .C=A. (B.C)

##### Distributive laws:

Law 1: A (B+C) =AB+AC

Law 2: A+BC= (A+B) (A+C)

##### Redundant literal rule (RLR):

Law 1: $A+\overline{A}B=A+B$

Law 2: $A(\overline{A}+B)=AB$

Law 1: A.A=A

Law 2: A+A=A

Law 1: A+A.B=A

Law 2: A (A+B)=A

#### Theorems of Boolean algebra:

##### Consensus theorem:

Theorem 1: $AB+\overline{A}C+BC=AB+\overline{A}C$

Theorem 2: $(A+B)(\overline{A}+C)(B+C)=(A+B)(\overline{A}+C)$

##### Transposition theorem:

Theorem: $AB+\overline{A}C=(A+C)(\overline{A}+B)$

##### De Morgan’s theorem:

Law 1: $\overline{A+B}=\overline{A}.\overline{B}$

Law 2: $\overline{AB}=\overline{A}+\overline{B}$

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