De-Morgan's Theorem | Digital Circuits - Electronics and Communication Engineering (ECE) PDF Download

What is De-Morgan's Theorem?

The rules of De-Morgan's theorem are produced from the Boolean expressions for OR, AND, and NOT using two input variables x and y. The first theorem of De-morgan's says that if we perform the AND operation of two input variables and then perform the NOT operation of the result, the result will be the same as the OR operation of the complement of that variable. The second theorem of De-Morgan says that if we perform the OR operation of two input variables and then perform the NOT operation of the result, the result will be the same as the AND operation of the complement of that variable.

De-Morgan's First Theorem

De-Morgan's Second Theorem

Let's take some examples in which we take some expressions and apply DeMorgan's theorems.
Example 1: (A . B . C)'
(A . B . C)' = A' + B' + C'

Example 2: (A + B + C)'
(A + B + C)' = A' . B' . C

Example 3: ((A + BC')' + D( E+ F')')'
For applying the DeMorgan's theorem on this expression, we have to follow the following expressions:
(1) In complete expression, first, we find those terms on which we can apply the De-Morgan's theorem and treat each term as a single variable.
So,
(2) Next, we apply De-Morgan's first theorem. So,
(3) Next, we use rule number 9, i.e., (A=(A')') for canceling the double bars.
(4) Next, we apply DeMorgan's second theorem. So,
(5) Again apply rule number 9 to cancel the double bar
Now, this expression has no term in which we can apply any rule or theorem. So, this is the final expression.

Example: (AB'.(A + C))'+ A'B.(A + B + C')'