## Introduction to Boolean Algebra

Boolean algebra which deals with two-valued (true / false or 1 and 0) variables and functions find its use in modern digital computers since they too use two-level systems called binary systems.

Let us examine the following statement:"I will buy a car If I get a salary increase **or** I win the lottery." This statement explains the fact that the proposition "buy a car" depends on two other propositions "get a salary increase" and "win the lottery". Any of these propositions can be either true or false hence the table of all possible situations:

Salary Increase |
Win Lottery |
Buy a car = Salary Increase **or** Win Lottery |

False |
False |
False |

False |
True |
True |

True |
False |
True |

True |
True |
True |

The mathematician George Boole, hence the name Boolean algebra, used 1 for true, 0 for false and + for the **or** connective to write simpler tables. Let X = "get a salary increase", Y = "win the lottery" and F = "buy a car". The above table can be written in much simpler form as shown below and it defines the OR function.

X |
Y |
F = X + Y |

0 |
0 |
0 |

0 |
1 |
1 |

1 |
0 |
1 |

1 |
1 |
1 |

Let us now examine the following statement:"I will be able to read e-books online if I buy a computer **and** get an internet connection." The proposition "read e-books" depends on two other propositions "buy a computer" and "get an internet connection". Again using 1 for True, 0 for False, F = "read e-books" , X = "buy a computer", Y = "get an internet connection" and use ^{.} for the connective **and**, we can write all possible situations using Boolean algebra as shown below. The above table can be written in much simpler form as shown below and it defines the AND function.

X |
Y |
F = X ^{.} Y |

0 |
0 |
0 |

0 |
1 |
0 |

1 |
0 |
0 |

1 |
1 |
1 |

We have so far defined two operators: **OR** written as + and **AND** written ^{.} . The third operator in Boolean algebra is the NOT operator which inverts the input. Whose table is given below where NOT X is written as X'.

The 3 operators are the basic operators used in Boolean algebra and from which more complicated Boolean expressions may be written. Example: F = X ^{.} (Y + Z)

## Rules and Theorems in Boolean Algebra

X + X = X

X ^{.} X = X

(X')' = X

X + X' = 1

X ^{.} X' = 0

X + Y = Y + X , X ^{.} Y = Y ^{.} X : commutative

X + (Y + Z) = (X + Y) + Z , X ^{.} (Y ^{.} Z) = (X ^{.} Y) ^{.} Z : associative

X ^{.} (Y + Z) = X ^{.} Y + X ^{.} Z

DeMorgan's Theorem

(x + Y)' = X' ^{.} Y'

(x ^{.} Y) = X' + Y'

## References and Books