## De Morgan’s Theorem

A mathematician named DeMorgan developed two theorems in the law of algebraic expression named” De Morgan’s Theorem. It use graphical symbol known as circuit to prove these theorem. These theorems are used in today world for developing many electronic circuit. Every electronic devices are firstly developed by using simple figure in the form of these theorem for input and output. This Theorem is further divided in 2 sub theorem:

- De Morgan’s First Theorem
- De Morgan’s Second Theorem

Proof for the both theorems are listed below.

## First De Morgan’s Theorem

The De Morgan’s fist theorem states thats, ” The complement of a sum equal to the product of the complements”.

Algebraic expression for the 1st theorem i.e (X+Y)’=X’.Y’

**Proof:**

**Graphical Symbol:**

** **** X’.Y’=(X.Y)’**

**Truth Table:**

Inputs | Output 1 | Output 2 | ||||
---|---|---|---|---|---|---|

X | Y | X+Y | (X+Y)’ | X’ | Y’ | X’.Y’ |

0 | 0 | 0 | 1 | 1 | 1 | 1 |

0 | 1 | 1 | 0 | 1 | 0 | 0 |

1 | 0 | 1 | 0 | 0 | 1 | 0 |

1 | 1 | 1 | 0 | 0 | 0 | 0 |

Comparing the values of (X+Y)’ and X’.Y’ from the truth table, both are equal, hence proved.**Conclusion:**

## Second De Morgan’s Theorem

De Morgan’s second theorem states , ” The Compliment of a product is equal to the sum of complements”.

Algebraic expression for 2nd theorem i.e (X.Y)’=X’+Y’

**Graphical Symbol**

**Truth Table:**

Inputs | Output 1 | Output 2 | ||||
---|---|---|---|---|---|---|

X | Y | XY | (XY)’ | X’ | Y’ | X’+Y’ |

0 | 0 | 0 | 1 | 1 | 1 | 1 |

0 | 1 | 0 | 1 | 1 | 0 | 1 |

1 | 0 | 0 | 1 | 0 | 1 | 1 |

1 | 1 | 1 | 0 | 0 | 0 | 0 |

By the help of these theorem in we can prove De Morgan’s Theorem. Making or designing software or circuit is very much easier using two theorems.

**Conclusion: **Comparing the values of (X.Y)’ and X’+Y’ from the truth table both are equal.