Content:
- Introduction
- Logical Operation
- Introduction
- Logical Operators
- Evaluation of Boolean Expression using Truth Table
- Basic Theorems of Boolean Algebra
- Principal of Duality
- DeMorgan's Theorems
- Logic Gates
- Logic Circuits
Introduction:
Boolean Algebra enables us to solve Logical problem i.e. problems whose answer will either be True or False. Boolean Algebra was created by George Boole to solve Logical problems mathematically. Claude E. Shannon used boolean algebra to solve logical problems on machine(Relay Switching Circuits). Now as computers deals with Logic problems, hence understanding and efficient use of Boolean Algebra leads to efficient circuits design which leads to cheaper and efficient computers.
Terminology:- Logical Statement: A statement which can either result in TRUE or FALSE and nothing else is called a logical statement
- Boolean Variable: This variable can have either TRUE or FALSE as value
- Truth Value: The value True and False are called Truth values.
Logical Operation:
Logical Function/ Statement:Logical statement/function are obtained by the combination of boolean variable and logical operator like AND, OR, NOT
Example:
Today its raining AND today its sunday
can be written as:
X AND Y
Logical Operator:
Terminology:
- Truth Table:A truth Table is a table which represents all the possible values of the logical variable/Statement along with all the possible result of the given combination of values.
- Tautology:If the result of any logical statement or expression is allways TRUE or 1 for all input combination, it is called Tautology.
- Fallacy:If the result of any logical statement or expression is always False or 0 for all input combination, it is called Fallacy
This operator operates on single variable and operator perform by NOT operator is called complement and the symbol used is bar. The truth table for not is listed below
OR Operatror:
OR operator denotes logical addition and '+' symbol is used to denote it. Thus A+B will be read as A OR B. The truth table is Listed below.
AND Operator:
AND operator is used to perform Logical multiplication in boolean algebra. The symbol used for AND operation is '.' dot. Thus A.B will be read as A AND B.
Evaluation of Boolean Expression using Truth Table:
Truth table can be used to verify the equality of LHS and RHS of a boolean expression. This can help in circuit minimization.
Example:
Practice:
- Verify using truth table (X+Y)'=X'Y' for each X,Y in (0,1)
- Prepare a table of combination for the following boolean logic expression
- X'Y'+X'Y
- XYZ'+X'Y'Z
- X'YZ'+X'Y'
Basic Theorems of Boolean Algebra:
Principal of Duality:
According to this principle, if we have postulates or theorems of Boolean Algebra for one type of operation then that operation can be converted into another type of operation (i.e., AND can be converted to OR and vice-versa) just by interchanging '0 with 1', '1 with 0', '(+) sign with (.) sign' and '(.) sign with (+) sign'. This principle ensures if a theorem is proved using postulates of Boolean algebra, then the dual of this theorem automatically holds and we need not prove it again separately.
Example:
DeMorgan's Theorems:
De Morgan has suggested two theorems which are extremely useful in Boolean Algebra.The theorem are given below:
Logic Gates:
A gate is a basic electronic circuit which operates on one or more signals to produce an output signal.
Gate are often called logic circuits because they can be analyzed with boolean logic. There are three types of logic gates:
- NOT
- OR
- AND
NOT Gate:
An Inverter(Not Gate) is a gate with only one input signal and one output signal; the output is always the opposite of the input state.
Logic Diagram
Truth Table:
OR Gate:
The OR Gate has two or more input but only one output signal. If any of the input signals is 1 (high), the output signal is 1(high).If all the input are 0 then output is also 0.
Logic Diagram:
Truth Table:
AND Gate:
The AND Gate can have two or more than two input signals and produce an output signal. When all the inputs are 1 i.e. high the the output is 1 otherwise output is 0.
Logic Symbol:
Truth Table:
Logic Circuits:
Optimized Booleans expression can be converted into circuits. As boolean expression can be used to implement logic and they can be optimized hence designing circuits from optimized expression makes the circuit cheaper and effective.
Example: Design a circuit to realise the following : AB + BC(B + C)
Practice:
Design circuit for the following boolean expression
- F(a,b,c) = AB + AC' + B'A'C
- F(X,Y,Z) = (X + Y) . (X' + Z') . (Y + Z')
Refrences:
- Computer Science with python by Sumita Arora