De Morgan’s Law

What is De Morgan’s Law

De Morgan’s law is the law that gives the relation between union, intersection, and complements in set theory. In Boolean algebra, it gives the relation between AND, OR, and complements of the variable, and in logic, it gives the relation between AND, OR, or Negation of the statement. With the help of De Morgan’s Law, we can optimize various boolean circuits involving logic gates which help us perform the same operation but with very few apparatus....

De Morgan’s Law in Set Theory

De Morgan’s law in the set theory defines the relationship between the union, intersection, and complements of the sets, and is given for both complement of union and intersection of two sets. In set theory, there are two De Morgan’s Laws that are:...

First De Morgan’s Law

First De Morgan’s law states that “The complement of the union of two sets is equal to the intersection of the complements of each set.”...

Second De Morgan’s Law

Second De Morgan’s law states that “The complement of intersection of two sets is equal to the union of the complements of each set.”...

Proof Using Algebra of Sets

Second De Morgan’s law: (A ∩ B)’ = A’ ∪ B’...

De Morgan’s Law in Boolean Algebra

De Morgan’s Law Boolean Algebra defines the relation between the OR, AND, and the complements of variables, and is given for both the complement of AND and OR of two values. In Boolean Algebra there are two De Morgan’s Laws that are:...

First De Morgan’s Law in Boolean Algebra

First De Morgan’s law states that “The complement of OR of two or more variables is equal to the AND of the complement of each variable.”...

Second De Morgan’s Law in Boolean Algebra

Second De Morgan’s law states that “The complement of AND of two or more variables is equal to the OR of the complement of each variable.”...

De Morgan’s Law Formula

Let’s compile all the formulas for De Morgan’s Law, in the following list....

Solved Examples on De Morgan’s Law

Problem 1: Given that U = {2, 3, 7, 8, 9}, A = {2, 7} and B = {2, 3, 9}. Prove De Morgan’s Second Law....

Showcase Examples of De Morgan’s Law

Context Example Logic Puzzles Puzzle: If it is not true that “It is raining and cold,” what can we infer? Application of De Morgan’s Law: We can infer that “It is not raining or it is not cold.” This uses De Morgan’s Law to simplify the negation of a conjunction into a disjunction. Programming Scenario: Checking if a number is neither positive nor even in a programming language. Code Snippet (Pseudocode): if !(number > 0 and number % 2 == 0) can be simplified using De Morgan’s Law to if (number <= 0 or number % 2 != 0). This demonstrates how De Morgan’s Law helps in simplifying conditional statements. Mathematical Proofs Statement: Prove that the complement of the intersection of two sets A and B is equal to the union of their complements. Application of De Morgan’s Law: According to De Morgan’s Law, (A ∩ B)’ = A’ ∪ B’. This shows how De Morgan’s Law is used to simplify expressions in set theory....

De Morgan’s Law Practical Examples

Example 1: Pizza Toppings...

Logic Applications of De Morgan’s Law

Application Area Description Logical Reasoning In logical puzzles or arguments, De Morgan’s Law helps simplify complex negations. For instance, negating “All apples are red” to “Not all apples are red” implies “Some apples are not red.” Computer Science De Morgan’s Law is crucial in optimizing conditional statements in programming. It allows programmers to simplify complex logical conditions, making code more efficient and readable. Electronic Circuit Design In digital electronics, De Morgan’s Law is used to design and simplify circuits. For example, it helps in converting AND gates into OR gates (and vice versa) using NOT gates, facilitating the creation of more efficient circuit layouts....

De Morgan’s Law – FAQs

State De Morgan’s first law statement in set theory....