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De Morgan's First Law

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Last updated date: 23rd Jul 2024
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What is De Morgan’s First Law?

The students at Vedantu will learn about the concept of De Morgan’s First Law in detail. They are provided with an entire guide to the topic to help them learn the basic concepts in depth. The website of Vedantu provides the students with lots of interesting and engaging study material including the concepts from Classes 1 to 12, revision notes, worksheets, sample papers, previous year’s question papers, NCERT Exemplar, and NCERT textbooks along with their solutions to clear the doubts of the students. Moreover, the students can refer to them easily without any hassle for free of cost at the time of their convenience. 

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In algebra, De Morgan's First Law or First Condition states that the complement of the product of two variables is corresponding to the sum of the complement of each variable. In other words, according to De-Morgan's first Laws or first theorem if ‘A’ and ‘B’ are the two variables or Boolean numbers. This indicates that the NAND gate function is similar to the OR gate function with complemented inputs. Then accordingly the equations are as below;-

For NOR Gate

Y=A+B=Y=A¯+B¯

For the Bubbled AND Gate

Y=A.B=Y=A¯.B¯

Symbolic representation of De Morgan's First Law Theorem

Since the NOR and the bubbled gates can be interchanged, i.e., both gates have just similar outputs for the identical set of inputs.

Hence, the equation can be algebraically represented in the figure shown below.

This equation presented above is known as DeMorgan's First Theorem. The symbolic illustration of the theorem is presented as shown below.

 

Role of Complementation Bars

Complementation bars are proposed to operate as grouping symbols. Hence, when a bar is broken, the expressions beneath it should remain grouped. Parentheses may be positioned around these grouped expressions as assistance to give a miss to changing precedence.

 

Verifying DeMorgan’s First Theorem Using Truth Table

According to DeMorgan's First Law, it proves that in conditions where two (or more) input variables are Added and negated, they are equal to the OR of the complements of the separate variables. Hence, the equivalent of the NAND function and is a negative-OR function verifying that A.B = A+B and we can prove this using the following table.

 

DeMorgan’s First Theorem Proof using Truth Table

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

Now that you have already understood DeMorgan's First Theorem using the Truth Table. We will make you familiar with another way to prove the theorem i.e. by using logic gates.

This is to say, we can also prove that A.B = A+B using logic gates as hereinafter.

 

Verifying and Execution of DeMorgan’s First Law using Logic Gates

The uppermost logic gate placement of: A.B can be executed considering a NAND gate with inputs A and B. The lowermost logic gate placement, in the beginning, inverts the two inputs yielding A and B which become the inputs to the OR gate. Thus, with this, the output from the OR gate becomes A+B.

Therefore, an OR gate with inverters (NOT gates) on its every input is equal to a NAND gate function, and an independent NAND gate can be showcased in this way the equality of a NAND gate is a negative - OR.

 

Simplifying DeMorgan’s First Law with Example

According to DeMorgan’s First Law, what is an equivalent statement to "The kitchen floor needs mopping and the utensils need washing, but I will not do both."?

The two postulations are "I will mop the kitchen floor" and "I will wash the utensils." Simply modify the given statement to an "OR" statement and negate each of these postulations:

"Either I will not mop the kitchen floor or I will not wash the utensils."

P.S: that this statement lays open the likelihood that one of the tasks is completed, and it is also possible that neither chore is being completed.

 

Importance of De Morgan's Law:

De Morgan’s first Law holds an important place in the syllabus of the students of higher secondary Classes, as it checks on a few crucial topics, it’s important features are highlighted below as:

  • The theorems of De Morgan's Law have been proved to be very useful for simplifying Boolean logic expressions due to the way they can 'break' an inversion, which could be the complement of a complex Boolean expression.

  • The theorems of De Morgan's can also be used to express logic expressions that do not originally contain inversion terms differently. Also, this can again prove to be useful while simplifying the Boolean equations. When used in this way it must be taken care of to not to ‘forget’ the final inversion, which can be easily avoided by complementing both the sides of the expression to be simplified before applying De Morgan's theorem and then again complementing after simplification.

  • Lastly, the students must note that one way of interpreting De Morgan's theorem is that any AND/OR operation can be considered as an OR/AND operation as long as NOT gates are used as well in the equation for ease of calculation.

 

Solved Examples

Problem1:

How to deduce the following equation to standard form?

F = MNO +M'N

F’ = (MNO + M’N)’l

Solution1:

Using the De Morgan's Law

We get,

= (MNO)’ (M’N)’

= (M’+N’+O’) (M+N’)

Now, applying the Law of distributivity

= N’ + (M’+O’) M

Again, applying Distributivity

= N’ + M’M + OEM

= N’ + MO’ (standard form)l

Problem2:

Apply De Morgan's Law to determine the inverse of the below given equation and reduce to the form of the sum-of-product:

F = MO' + MNP' + MOP

Solution2:

F’= (MO' + MNP' + MOP)’

= (MO’)’ (MNP’)’ (MOP)’

= (M’+O)(M’+N’+P)(M’+O’+P’)

= M’+O (N’+P) (O’+P’)

= M’+ (N’+P) OP’

= M’ + ON’P’ + OPP’

Thus, we get = M’ + ON’P.

 

Fun Facts

  • Do you know the full form of DeMorgan’s Theorems? It’s Demorgan’s theorem.

  • No matter whether De Morgan's Laws apply to sets, propositions, or logic gates, the anatomy always remains the same.

FAQs on De Morgan's First Law

1. What are DeMorgan's Theorems?

DeMorgan’s Theorems typically explain the correspondence between GATES with inverted inputs and GATES with inverted outputs. In simple terms, a NAND gate equals a Negative-OR gate, while a NOR gate equals a Negative-AND gate. There are 2 DeMorgan’s Theorems i.e.

  1. DeMorgan’s First Law or Theorem

  2. DeMorgan’s Second Law or Theorem

When “breaking-up” a complementation bar in a Boolean expression or equation, the operation without any deviation beneath the break (addition or multiplication) overturns, and moreover the bits of the broken bar still remain over the respective terms or variables.

Time and again, it becomes easier to deal with a Mathematical problem by fragmenting the longest (uppermost) bar before breaking any bars beneath it. However, you should never try to break two bars in one step!

2. Why is DeMorgan’s Theorem Useful?

DeMorgan’s Theorem is chiefly used to solve the different and the longest Boolean algebra expressions. As mentioned above, this theorem describes the equality between the gate with identical inverted input and inverted output thus making it a common application for incorporating the fundamental gate operations like NAND gate and NOR gate. Various other uses of DeMorgan’s Theorem include:-

  • It is most widely executed in digital programming and even for drawing digital circuit diagrams.

  • This Law is also applicable in computer engineering to create logic gates.

  • It essentially explains how Mathematical concepts, statements as well as expressions are linked through their opposites.

  • In set theory, the theorem relates to the union and bisection of sets through complements.

  • In propositional rationale, De Morgan's theorems establish a link between conjunctions and disjunctions of propositions by way of negation.

3. How is De Morgan’s first Law used?

We use De Morgan's Laws usually to write a statement that must be equivalent to the following statement: "It is not true that North Dakota and East Dakota are both states." North Dakota is not a state, and East Dakota is not a state. Either East Dakota is a state, or North Dakota is not a state. This way the Law helps to get an answer for the statement as to which of these is a state and which one is not. Moreover, it is important in many other ways, apart from the example quoted above.

4. How can I use De Morgan's Law to negate a statement?

The students can use De Morgan’s Law to negate a statement in a way that is defined here as to negate an “and” statement, the student needs to negate each part and change the “and” to “or”. Since the negation of a disjunction is equivalent to the conjunction of the negation of the statements making up the disjunction. For negating an “or” statement, the students need to negate each part and change the “or” to “and”. This way the students can easily negate the statements.

5. Where Can I get De Morgan’s First Law?

The students can get De Morgan’s first Law concert and details in the form of a proper guide at Vedantu’s website for free. The topic has been explained in detail with the definition, importance, formulas, uses, and other relevant details. Since Vedantu has an objective of helping the students improve their performance there are several other resources available on the website in the form of worksheets, notes, sample papers, syllabus, previous year’s question papers, and others such to get a clear understanding of the concept and inculcate the same in their learning well.