# Postulates and Theorems of Boolean Algebra

Boolean algebra is a system of mathematical logic, introduced by a mathematician George Boole in 1854. Boolean algebra differs from ordinary algebra and binary number system. It is a binary algebra defined to perform binary and logical operations.

Take a look at some of the terminologies used in Boolean Algebra, different postulates and theorems of Boolean algebra, which forms the basics for designing a combinational circuits.

## Terminologies used in boolean Algebra

** Variable **– The symbol which represent an arbitrary elements of an Boolean algebra is known as Boolean variable. In an expression, Y=A+BC, the variables are A, B, C, which can value either 0 or 1.

** Constant **– It is a fixed value. In an expression, Y=A+1, A represents a variable and 1 is a fixed value, which is termed as a constant.

* Literal *– Each occurrence of a variable in Boolean function either in complemented or normal form is said to be literal.

## Postulates of Boolean Algebra

S.No. | Name of the Postulates | Postulate Equation |
---|---|---|

1 | Law of Identity | A + 0 = 0 + A = A A . 1 = 1 . A = A |

2 | Commutative Law | (A + B) = (B + A) (A . B) = (B . A) |

3 | Distributive Law | A . (B + C) = (A . B) + (B . C) A + (B . C) = (A + B) . (B + C) |

4 | Associative Law | A + (B + C) = (A + B) + C (A . B) . C = A . (B . C) |

5 | Complement Law | A + A’ = 1 A . A’ = 0 |

## Theorems of Boolean Algebra

S.No | Theorem | Statement | Equations |
---|---|---|---|

1 | Duality Theorem | A boolean relation can be derived from another boolean relation by changing OR sign to AND sign and vice versa and complementing the 0s and 1s. | A + A’ = 1 and A . A’ = 0 are the dual relations. |

2 | DeMorgan’s Theorem 1 | Complement of a product is equal to the sum of its complement. | (A . B)’ = A’ + B’ |

3 | DeMorgan’s Theorem 2 | Complement of a sum is equal to the product of the complement. | (A + B)’ = A’ . B’ |

4 | Idempotency Theorem | – | A + A = A A . A = A |

5 | Involution Theorem | – | A” = A |

6 | Absorption Theorem | – | A + (A . B) = A A . (A + B) = A |

7 | Associative Theorem | – | A + (B + C) = (A + B) + C A . (B . C) = (A . B) . C |

8 | Consensus Theorem | – | AB + A’C + BC = AB + A’C (A + B) + (A’ + C) + (B + C) = (A + B) + (A’ + C) |

9 | Uniting Theorem | – | AB + AB’ = A (A+B) + (A + B’) = A |

10 | Other theorems | – | A + 1 = 1 A . 0 = 0 |

11 | Other theorems | – | A + (A’ . B) = A + B A . (A’ + B) = A . B |

## Solved Examples

Let us solve some examples of boolean function by applying the postulates and theorems of boolean algebra.

**Simplify A . (AB + C)**

**Simplify A + A’B**

**Simplify Y = AB’D + AB’D’**

**Simplify Y = (AB’ (C+BD) + A’B’)C**

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