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Boolean Algebra - Basics

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Boolean algebra is branch of mathematics which came into picture in 1854 with George Boole's revolutionary paper "An Investigation of the Law of Thought". Basically, it is logic based algebra involved with binary number system. Practically, it can be applied to any decision making situation. Presently, it is widely used in computer electronics.


Concept

Binary number system consists of only 1's and 0's. 1 represents truth and 0 represents false. Any variable which stores these truth values are logical variables. Any operator in Boolean algebra can be explained using truth table. Truth table represents all the possible values of logical variables along with all the possible outcomes of given combination of values.


Operators in Boolean Algebra

AND OPERATOR: This operator is used when final decision (output) depends equally on both factors (input). The AND operator is represented with help of '.' or '^' AND operator is also known by the name conjugation.

And Operator
NOTE: The AND operator is true only when both the inputs are true.


OR OPERATOR: This operator is used when final decision (output) depends on either of one factor (input). The OR operator is represented by '+'. It is also known by name dis-junction.

Smiley face
NOTE: The OR operator is true when any one of the input is true.


NOT OPERATOR: This operator is used to invert the input. This operation is also called complementation or negation. The NOT operator is represented with help of '`' or using bar '-' above the logic variable.

Smiley face


PRINCIPLE OF DUALITY:

The principle of duality states that any Boolean expression is equivalent to the Boolean expression obtained by replacing OR operator with AND operator, AND operator with OR operator, 1's with 0's and 0's with 1's.

BOOLEAN EXPRESSION NAME OF LAW
0+X=0 PROPERTY OF 0
0.X=0 PROPERY OF 0
1+X=1 PROPERTY OF 1
1.X=X PROPERTY OF 1
X+X=X INDEMPOTENCE LAW
X.X=X INDEMPOTENCE LAW
X``=X INVOLUTION
X+X`=1 COMPLEMENTARITY LAW
X.X`=0 COMPLEMENTARITY LAW
X+Y=Y+X COMMUTATIVE LAW
X.Y=Y.X COMMUTATIVE LAW
X+(Y+Z)=(X+Y)+Z ASSOCIATIVE LAW
X.(Y.Z)=(X.Y).Z ASSOCIATIVE LAW
X(Y+Z)=XY+XZ DISTRIBUTIVE LAW
X+YZ=(X+Y)(X+Z) DISTRIBUTIVE LAW
X+XY=X ABSORPTION
X.(X+Y)=X ABSORPTION
(X+Y)`=X`.Y` DEMORGAN'S THEOREM
(X.Y)`=X`+Y` DEMORGAN'S THEOREM
X+X`Y=X+Y -



Demorgan's Theorems

All the above laws can be verified using truth table. Also, any Boolean expression can be reduced using above law. It is called algebraic method of reduction.


DEMORGAN'S FIRST THEOREM: (X+Y)`=X`.Y`
PROOF:



DEMORGAN'S SECOND THEOREM: (X.Y)`= X`+Y`
PROOF:



APPLICATIONS:

  • Can be applied to any two valued situation. When we have to make any decision in our day to day life, we always apply the basics of Boolean algebra. Depending on the importance of the each factor involved, the final decision taken by us always follows the laws of Boolean algebra.

  • Used in switching circuit. Boolean algebra has an important role in circuits of switches. At a very basic level, two switches connected in series is analogous to AND operator and the same switches connected in parallel is analogous to OR operator. Further, any complicated switch circuit can be simplified by reducing the equivalent Boolean expression.

  • Used in set theory. We have terms like union and intersection in set theory which are equivalent to AND and OR in Boolean algebra. DeMorgan's laws are also true here. In mathematics, set theory is used for various statistical calculations.

  • Used in digital electronics. In digital electronics, a +5 volt is considered HIGH(1) and 0 volt as LOW(0). The logic for various digital circuits like seven segment display, decoder, adder, etc is developed using Boolean algebra.







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