- Definition of De Morgan’s Theorem: De Morgan’s theorem is defined as a set of rules used to transform and simplify logical expressions in digital circuit design.
- Simplification of Logical Expressions: By using De Morgan’s theorem, conjunctions and disjunctions can be expressed in terms of each other through negation, simplifying complex logic.
- First Theorem: The first theorem states that the complement of the product of two variables equals the sum of their complements.
- Second Theorem: The second theorem states that the complement of the sum of two variables equals the product of their complements.
- Application in Digital Electronics: De Morgan’s theorem is widely used in Boolean algebra and digital electronics to simplify circuit designs and calculations.
What are De Morgan’s Laws?
De Morgan’s laws (also known as De Morgan’s theorem) are a pair of transformation rules used to simplify logical expressions in computer programs and digital circuit designs. They are named after their founder Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.
This might sound like complicated math at first.
So let’s express these rules in English:
- the negation of a disjunction is the conjunction of the negations
- the negation of a conjunction is the disjunction of the negations
Or even more plainly as:
- the complement of the union of two sets is the same as the intersection of their complements
- the complement of the intersection of two sets is the same as the union of their complements
When combined with binary arithmetic (binary addition, binary subtraction, binary multiplication and binary division), De Morgan’s Laws help simplify logical circuits.
Before discussing De-Morgan’s theorems, we should know about compliments.
Complements are the reverse value of the existing value. We are trying to say that there are only two digits in the binary number system 0 and 1. If A = 0, then the complement of A will be 1 or A’ = 1.
De Morgan’s Laws
De Morgan proposed two main theorems. These theorems help solve many Boolean algebra problems, which are crucial in digital electronics. De Morgan’s theorem states:
Theorem 1:
The complement of the product of two variables is equal to the sum of the complement of each variable.
Thus according to De-Morgan’s laws or De-Morgan’s theorem, if A and B are the two variables or Boolean numbers. Then accordingly
Theorem 2:
The complement of the sum of two variables is equal to the product of the complement of each variable.
Thus according to De Morgan’s theorem, if A and B are the two variables, then.
De-Morgan’s laws can also be implemented in Boolean algebra in the following steps:-
- While doing Boolean algebra at first replace the given operator. That is, if (+) is there, then replace it with (.), and if (.) is there, then replace it with (+).
- Find the next complement of each of the terms.
De-Morgan’s theorem can be proved by the simple induction method from the table given below:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| A | B | A’ | B | A+B | A.B | (A+B)’ | A’.B’ | (A.B)’ | A’+B’ |
| 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
Now, look at the table very carefully in each row.
Firstly the value of A = 0 and the value of B = 0. Now for this values A’ = 1, B’ = 1. Again A + B = 0 and A.B = 0. Thus (A + B)’ = 1 and (A.B)’ = 1, A’ + B’ = 1 and A’.B’ = 1.
From this table, you can see that the value of columns no 7 and 8 are equal, and columns no 9 and 10 are also equal, proving De-Morgan’s theorem.
Again different values of A and B we see the same thing, i.e. column no 7 and 8 are equal to each other and 9 and 10 are equal to each other. Thus by this truth table, we can prove De-Morgan’s theorem.
Some examples given below can make your idea clear.
Therefore, 
With the help of De-Morgan’s theorem, our calculation becomes much easier.
Let another example be,
In both the equations, we have suitably used De-Morgan’s laws to make our calculation much easier.
