Applications of logic

1

History of Logic

In the year 1847, English mathematician George Boole (1815 - 1864)

published, 'The Mathematical Analysis of Logic'. This book of his showed how

using a specific set of logic can help one to wade through piles of data to find

the required information. The importance of Boole's work was his way of

approach towards logic. By incorporating it into mathematics, Boole was able

to determine what formed the base of Boolean algebra. It was the analogy

which algebraic symbols had with those that represented logical forms. This

basic analogy gave birth to what is known as the Boolean algebra. As we know

that working of computers are based on the binary number system (1 or 0),

where ‘1’ means 'ON' and ‘0’ signifies 'OFF'. These two states are

represented by a difference in voltage. Now, the application of this system to

the computer's binary number system was incorporated by an MIT grade

student Claud Shannon. This was how the Boolean search came into place.

The Symbols

Precisely, this system is defined as a logical system of operators –

'AND’

'OR'

'NOT'

and is a way of comparing individual bits. These connectors or operators are

now used in computer construction, switching circuits, etc.The AND, OR, and

NOT operators are also known as logic gates, and are used in logical

operation. Their schematic diagram can be viewed from any book based on

Applications of logic

2

Boolean algebra. The following paragraphs describe the symbols and the

operation.

The AND Gate

The AND gate is denoted by a dot

(.). In an AND gate, there will be

more than one input and only one

output. Here, if all inputs are ON,

the output will also be ON. And, if either of the

inputs is OFF, then the output will also be OFF.

The AND gate's symbol is '&'. Let's see the

working in an example.

A . B = C (Here, A and B are the inputs, and C is

the output)As we know that in the binary number

system, 1 means ON and 0 means OFF. So, if we

take the inputs to be 1, the output will also give us

1.

A . B = C

1 . 1= 1 (A = 1, B = 1).

If any of the input is taken as 0, then output will also be 0

A . B = C

1 . 0 = 0 (A = 1, B = 0)

The OR Gate

The OR gate is denoted by plus

(+). Here, there will be more

Applications of logic

3

than one input and just one output. If we take both the inputs as 1, the output will also be 1. However, unlike the AND gate, if either

of the inputs is 0, the output will still be one. Its symbol is '/'. Example;

A + B = C (Here, A and B are the inputs, and C is the output)

For A = 1, B = 1 A + B = C

1 +1 = 1

For A = 1, B = 0 A + B = C

1 + 0 = 1

For A = 0, B = 0 A + B = C 0 + 0 = 0

The NOT Gate The NOT Gate is also known as the inverter gate. As the name suggests, here

the output will be opposite to the

input. There will be one input and one output. That is, if the

input is 1 (ON), then the output will be 0 (OFF). The NOT

gate is symbolized by a line over top of the input (Ā). The sign is also known as a 'complement'. For example,

For example, if A is the input, the output will be Ā that is,

For A = 1, output is 0 And for A = 0, output is 1

The NAND and the NOR gates are known to be the universal gates. Their

combinations may be used to form any kind of logic gates. A NAND gate is

formed by combining a NOT and AND gate. A NOR gate is a combination of a

NOT and OR gate. The other gates are

Applications of logic

4

XOR (exclusive OR) and XNOR gates.

Combinational circuit

A combinational circuit is a compound circuit consisting of the basic logic

gates such as NOT, AND, OR.

Determining output for a given input

Indicate the output of the circuit below when the input signals are

P = 1, Q = 0 and R = 0

The out of this circuit ‘S’ is =1

Applications of logic

5

Labeling intermediate outputs

Boolean Expression

In this Expression trace the circuit from left to right and

write down the output of each logic gate

Applications of logic

6

And also give a Boolean expression and make a Circuit

Hence (P∨Q) ∧ (P∨R) is the Boolean expression for this

circuit.

Construct circuit for the Boolean expression (P∧Q) ∨ ~R