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# De Morgan's Theorem of Boolean Algebra

There are two theorems given by De Morgan's for boolean algebra that we are going to discuss in this article or tutorial.

## De Morgan's First Theorem

De Morgan's first theorem states thatĀ the complement of the sum equals the product of the complement.

### Boolean Expression of De Morgan's First Theorem

Here is the boolean expression of De Morgan's first theorem:

(A+B)' = (A'.B')

Here, A and B are the two binary variables. Binary variables are those in which both variables can hold either 0 or 1.

### Graphical Representation of De Morgan's First Theorem

Below is the symbol or graphical representation of De Morgan's first theorem:

As you can see from the above figure, the firstĀ gate is a NOR gate, which is taking two inputs, A and B, and giving the output Y that will
be equal to **(A+B)'**. And the second gate is the AND gate, but here there are two inputs A and B that are passing through a NOT gate and then
enter the AND gate as input. Therefore, at the AND gate's input terminal, both the inputs will become A' and B', and therefore the output Y holds
the value **(A'.B')**.

### Proof of De Morgan's First Theorem using the Truth Table

Here is the truth table that identifies the proof of De Morgan's first theorem equation, which is **(A+B)'=(A'.B')**:

A | B | A' | B' | A+B | (A+B)' | A'.B' |
---|---|---|---|---|---|---|

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

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

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

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

As you can clearly see, all the corresponding row's values of **(A+B)'** and **(A'.B')** are same. Therefore, the first theorem of De Morgan
is proved using the above truth table.

## De Morgan's Second Theorem

According to De Morgan's second theorem, the complement of produce equals the sum of the complement.

### Boolean Expression of De Morgan's Second Theorem

The boolean expression of De Morgan's second theorem is given below:

(A.B)' = (A'+B')

### Graphical Representation of De Morgan's Second Theorem

Here is the symbol or graphical representation of the second theorem given by De-Morgan:

Here, the first gate is a NAND gate; there are two inputs, A and B, provided here, which give Y at the output and will be equal to
**(A.B)'**. And the second gate is an OR gate, and the inputs A and B provided here are passed through a NOT gate; therefore, these two
inputs become A' and B' at output, which will be the input for the OR gate that gives Y at output that will be equal to **(A'+B')**.

### Proof of De Morgan's Second Theorem using Truth Table

Here is the truth table to demonstrate De Morgan's second theorem:

A | B | A' | B' | A.B | (A.B)' | A'+B' |
---|---|---|---|---|---|---|

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

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

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

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

Again, as you can see, **(A.B)'** holds the same value as **(A'+B')** holds in all corresponding rows. Therefore, De Morgan's second
theorem is demonstrated using the above truth table.

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