George Boole published The Mathematical Analysis of Logic in 1847, showing how logic could be used to organize data. This established the foundations of Boolean algebra and helped develop the digital logic now used in computers. Boolean algebra represents logical operations like AND, OR, and NOT using binary digits of 1 and 0. These logic gates were then incorporated into digital circuits by Claude Shannon and are now fundamental to computer hardware and switching circuits.

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History of Logic
In the year 1847, Englishmathematician George Boole (1815 -1864)
published, 'The MathematicalAnalysis of Logic'. This book of his showedhow
using a specific set of logic canhelp one to wade through piles of data to find
the required information. The importance of Boole's work was his way of
approachtowards 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 logicalforms. This
basic analogygave birth to what is knownas the Booleanalgebra. 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
representedby a difference in voltage. Now, the application of this systemto
the computer's binary number system was incorporatedby an MIT grade
student Claud Shannon. This was how the Booleansearchcame into place.
The Symbols
Precisely, this systemis defined as a logicalsystem of operators –
'AND’
'OR'
'NOT'
and is a way of comparing individual bits. These connectorsoroperators are
now used in computer construction, switching circuits, etc.The AND, OR, and
NOT operators are also known as logic gates, andare used in logical
operation. Their schematic diagram can be viewed from any book basedon

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Booleanalgebra. 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 takenas 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

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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 signis 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

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XOR (exclusive OR) and XNOR gates.
Combinational circuit
A combinational circuit is a compound circuit consisting of the basic logic
gates suchas 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

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Labeling intermediate outputs
Boolean Expression
In this Expression trace the circuit from left to right and
write down the output of each logic gate

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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