**Boolean Logics & Algebra**

Topics

Boolean logic: NOT, AND, OR, NAND, NOR, XOR, Truth table, De Morgan’s laws and logic circuits

**Logical Statements **–

The statement or sentences which has only two possible answers – yes or no, called Logical statements or truth function.

**Logical Variables / Boolean Variable :**

The variables which holds the logical value (constant) true or false; 1 or 0; yes or no; called Logical variable.

**Boolean Algebra **

Algebra of two values (true or false, yes or no, 1 or 0) called Boolean Algebra. It is a tool used an important tool in analyzing, designing and implementing digital circuits.

**Truth table**

Truth table is a table which represents all the possible values of logical variables / statements along with all the possible results of the given combination of values.

**Logical operators:**

Operators which are use to process logical values (1 or 0; true or false); called Logical operators. **NOT, OR & AND are the logical operators.**

NOT Operator | OR Operator | AND Operator |

This operator operates on single variable and the operation is called complementation. This operation inverts the value of the variable. | This operation is called logical addition and the symbol used is +. X+Y is read as X OR Y. The possible input and output combinations are given below:- 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1 | The AND operation is called logical multiplication. The . (dot) symbol is use to represent AND. X . Y is read as X AND Y. The possible input and output combinations are given below:- 0 . 0 = 0 0 . 1 = 0 1 . 0 = 0 1 . 1 = 1 |

**Truth table for NOT, OR & AND logical operators:**

NOT Operator | OR Operator | AND Operator |

x x’ 1 0 0 1 | x y x+y 0 0 0 0 1 1 1 0 1 1 1 1 | x y x.y 0 0 0 0 1 0 1 0 0 1 1 1 |

**Basic logic gates**

A gate is a basic electronic circuit which operates on one or more input signals to produce an output signal. There are eigth

types of logic gates. Which are categorised into three categories. These are-

- Basic Gates
- NOT Gate / INVERTER
- OR Gate
- AND Gate

- Universal Gates
- NOR Gate
- NAND Gate

- Special Gates
- XOR Gate
- XNOR Gate

**NOT gate (Inverter):**

An inverter is a gate with only one input signal and one output signal. The output state is always the opposite of the input state i.e. if input state is 1 then output state is 0, and if input state is 0 then output state is 1.

**OR gate:**

The OR gate has two or more input signals, but only one output signal. If all the inputs are 0 , then the output is 0. If one or more inputs are 1, then the output is 1.

**AND gate:**

The AND gate can have two or more input signals, but one output signal. If any of the input is 0, the output is 0. If all the inputs are 1, the output is 1.

**Universal Gates** : NOR and NAND are called universal gates. You can make basic gates (NOT, OR & AND) with the help of NOR and NAND gate, that’s why it is called Universal Gates.

**NOR gate:**

The NOR gate has two or more input signals, but only one output Signal. NOR gate is inverted OR gate. If all the inputs are 0, then the output is 1. If any of the input is 1,the output is 0.

**NAND gate:**

The NAND gate has two or more input signals, but only one output. NAND gate is inverted AND gate. If all the inputs are 1, then the output is 0, otherwise it produces output 1.

**Special Gates:****XOR gate(Exclusive OR gate)**

The XOR gate can have two or more inputs , but only one output. When there are odd number of 1’s in the input, then it produces output 1, otherwise output is 0.

**XNOR gate(Exclusive NOR gate)**

The XNOR gate can have two or more inputs , but only one output. When there are even number of 1’s in the input, then it produces output 1, otherwise output is 0.