# DeMorgan’s Theorem Proof with Examples and Tables

by Luzan Baral | May 9, 2012 | Digital Electronics, Engineering |

**DeMorgan’s Theorem:**

**Theorem 1:**

Statement: “The complement of two or more variable ANDed is equivalent to the OR of the Complements of the individual variables.

**(XY)’ = X’ + Y’**

DeMorgan’s Theorem 1

X |
Y |
(XY)’ |
X’+Y’ |

0 |
0 |
1 |
1 |

0 |
1 |
1 |
1 |

1 |
0 |
1 |
1 |

1 |
1 |
0 |
0 |

**Explanation**:

According to this theorem the result of the first NAND gate is equivalent to the result of the second Negative-OR gate when the X and Y is reversed and passed to the Negative-OR gate.

**Theorem 2**

Statement: “The complement of two or more variables ORed is equivalent to the AND of the complements of the individual variables”.

**(X+Y)’=X’Y’**

DeMorgan’s Theorem 2

X |
Y |
(X+Y)’ |
X’Y’ |

0 |
0 |
1 |
1 |

0 |
1 |
0 |
0 |

1 |
0 |
0 |
0 |

1 |
1 |
0 |
0 |

Explanation:

According to De Morgans theorem the result of the first NOR gate is equivalent to the result of the second Negative-AND gate when the X and Y is reversed and passed to the Negative-AND gate.

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