# Boolean Algebra - Basics

**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.

**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.

**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.

## 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:

**CACKL**

**E**comment system