Overview Of DeMorgan’s First Theorem

De Morgan’s Theorem is an especially useful technique in digital design. The complement of the product among all terms is equal to the total of the complements of each term, according to the theorem. Similarly, the product of the complements of each term equals the complement of the total of all terms. A NAND gate is identical to an OR gate having reversed inputs, according to De Morgan’s theorem. It could be used to show that an AND gate with reversed inputs is equivalent to a NOR gate In this article, you will learn in detail about De morgan’s theorems, and specifically the first theorem.  

De Morgan’s theorems are a set of transformation rules which are both valid rules of inference in propositional logic as well as Boolean algebra. The principles allow conjunctions and disjunctions to be expressed solely in terms of each other through negation.

The regulations are as follows:

The conjunctions of the negations would be the negation of the disjunctions

OR

The intersection of the complement of two sets is the complement of their respective union.

(neither A / B) = (not A) &(not B)

not (A & B) = (not A) / (not B), 

where “A or B” is an “inclusive or,” implying that at minimum one of A or B exists, instead of an “exclusive or,” implying that only one of A or B exists.

These are officially written as in set theory & Boolean algebra.

A B = A B

A B = A B

The two most essential theorems of boolean algebra were created by a great mathematician named DeMorgan. The De Morgan theorems are being used to prove that the NOR & negative-AND gates, as well as the negative-OR & NAND gates, are mathematically equivalent. These theorems are crucial in the solution of Boolean algebra expressions. The logical function for each set/combination of input variables is described in the table beneath.

The rules of De Morgan Theorems are generated utilising 2 input values x and y as well as Boolean formulas for OR, AND, and NOT. De morgan’s first theorem states that if we conduct the AND operation on 2 input variables and afterwards the NOT operations on the output, the outcome would be the same as the OR operation on that variable’s complement. The second DeMorgan theorem states that if we conduct the OR operation on 2 input variables and afterwards the NOT operation on the output, the outcome would be the same as the AND operation on that variable’s complement.

The First Theorem of De-Morgan

The complement outcome of the AND operation is identical to the OR operation of the complements of that variable, including the first theorem. As a result, it is equal to the NAND function as well as being a negative-OR function, demonstrating that (A.B)’ = A’+B’, as shown in the table below.

A NOR gate is equal to a bubbling AND gate, as per DeMorgan’s first theorem. The equation displayed beneath can be used to describe the bubbling AND gate’s Boolean expressions.

The formula for a NOR gate is:

Z = A +B

• Because the NOR or bubbled gates are convertible, i.e., both have completely identical outputs for the same set of inputs, the equation for the bubbled AND gate is as follows:

Z= A . B

A + B = A . B

DeMorgan’s Theorem is the name for equation 1 or identity presented above. The following diagram shows the conceptual model of the theorem:

The equivalence of gates using inverted inputs & gates having reversed outputs is described by DeMorgan’s Theorems. A NAND gate is the same as a Negative-OR gate, whereas a NOR gate is the same as a Negative-AND gate.

When a complimentary bar in a Boolean function is “broken,” the operation right beneath it (adding or multiplying) flips, and the broken bar pieces stay well over respective terms.

It’s usually better to approach a challenge by breaking the largest (uppermost) bar first, then the bars beneath it. It’s never a good idea to try to break two. in one stride!

Complementation bars serve as signals of grouping. As a result, if a bar is damaged, the phrases behind it must still be grouped. To prevent shifting precedence, parentheses can be used around all these combined terms.

Conclusion

To summarise, De Morgan’s theorems are a set of transformation rules which are both valid rules of inference in propositional logic as well as Boolean algebra. The principles allow conjunctions and disjunctions to be expressed solely in terms of each other through negation. De Morgan’s First Law, often known as the First Condition, says that the complement of the product of two variables equals the total of the complements of every variable in algebra. Alternatively,  ‘A’ & ‘B’ are 2 factors or Boolean numbers, as per De-Morgan’s Laws or first theorem.