De Morgan's Theorem

De Morgan's Theorem 1:

The complement of the sum of two or more variables is equal to the product of the complement of the variables.

De Morgan's Theorem 2:

The complement of the product of two or more variables is equal to the sum of the complements of the variables.

For two variables A and B these theorems are written in Boolean notation as follows

Demorgan's Theorem Equation 1Demorgan's Theorem Equation 2

The two theorems are proved below.

To prove                      

Demorgan's Theorem Equation 1

Since each variable can have a value either 0 or 1, the following four cases arise:

Demorgan's Theorem Prove Equation

Since, in every case the left hand side equals the right hand side, the theorem is proved.

To prove

Demorgan's Theorem Equation 2

Since each variable can have a value either 0 or 1, the following four cases arise:

Demorgan's Theorem Equation

As all possible combinations of A and B are exhausted, the theorem is proved.

Consensus:

Demorgan's Theorem Consensus Equation 1

Let us prove part (a) algebrically,

Demorgan's Theorem Consensus Equation 1 Prove

Part (b) is the dual of part (a) and can be similarly proved.

If we want to prove that (b) by truth table method, we can do so by enumerating all the eight possible combinations.

A B C A+B +C B+C LHR RHS
0 0 0 0 1 0 0 0
0 0 1 0 1 1 0 0
0 1 0 1 1 1 1 1
0 1 1 1 1 1 1 1
1 0 0 1 0 0 0 0
1 0 1 1 1 1 1 1
1 1 0 1 0 1 0 0
1 1 1 1 1 1 1 1