Anaya Nolan
De Morgan's laws are two sets of rules or laws developed from Boolean expressions for AND, OR, and NOT using two input variables. These laws, named after Augustus De Morgan, allow the input variables to be negated and converted from one form of a Boolean function into another. De Morgan's laws can be applied to the negation of a disjunction or the negation of a conjunction in all or part of a formula. For example, if you are told at a pizza party that you can choose any toppings except for both mushrooms and olives, De Morgan's Law states that you can either choose to not have mushrooms or choose to not have olives.
| Characteristics | Values |
|---|---|
| Named After | Augustus De Morgan |
| Date | 19th Century |
| Application | Boolean Algebra, Logic, Mathematics |
| Function | Finding Equivalency, Changing ORs to ANDs and Vice Versa |
| Use | Constructing Truth Tables, Simplifying Digital Circuits |
| Number of Laws/Theorems | Two |
| Logic Functions | AND, OR, NOT, NAND, NOR |
| Logic Gates | NOT, AND, OR, NAND, NOR, XOR, XNOR |
| Use in Sentences | Can be used to determine what is allowed and what is not allowed |
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What You'll Learn
De Morgan's theorem and its application to the negation of a disjunction
De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. De Morgan's formulation was influenced by the algebraization of logic undertaken by George Boole.
De Morgan's theorem can be applied to the negation of a disjunction or the negation of a conjunction in all or part of a formula. In the case of its application to a disjunction, consider the following claim: "It is false that either A or B is true". In symbolic logic, this can be written as "¬(A ∨ B)". This means that neither A nor B is true, so it must follow that both A and B are not true, which can be written as "(¬A) ∧ (¬B)". If either A or B were true, then the disjunction of A and B would be true, making its negation false.
The negation of this disjunction must thus be true, and the result is identical to the first claim. This follows the logic that "since two things are both false, it is also false that either of them is true". Working backward, the second expression asserts that A and B are both false (or that "not A" and "not B" are true). Knowing this, a disjunction of A and B must be false as well.
De Morgan's laws are used to define logical equivalences of a statement and can be very useful in simplifying complex logical expressions. For example, suppose we know that some statement of the form "~(P ∨ Q)" is true. De Morgan's laws tell us that "(~P) ∧ (~Q)" is also true, and hence that "~P" and "~Q" are both true as well.
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De Morgan's laws in Boolean algebra
De Morgan's laws, named after Augustus De Morgan, are a set of rules in logic and mathematics that establish the relationship between logical connectives and their negations. They are particularly useful in Boolean algebra, where they help define logical equivalences of statements.
The first of De Morgan's laws states that the negation of a conjunction is logically equivalent to the disjunction of the negations. In other words, if we have two statements, P and Q, the negation of "P and Q" is equivalent to "not P or not Q". This can be expressed in symbolic logic as:
~(P ∧ Q) ≡ ~P ∨ ~Q
The second of De Morgan's laws is related to the negation of a disjunction. It states that the negation of "P or Q" is equivalent to "not P and not Q". Symbolically, this can be written as:
~(P ∨ Q) ≡ ~P ∧ ~Q
De Morgan's laws can be applied to part of a formula or statement. For example, consider the statement "it is false that either A or B is true". By applying De Morgan's laws, we can conclude that "neither A nor B is true", which can be further simplified to "A is not true and B is not true". This illustrates how De Morgan's laws can be used to transform statements with complex negations into logically equivalent statements that are often simpler and easier to understand.
In Boolean algebra, De Morgan's laws have practical applications in circuit design and logic gates. For instance, an OR gate can be constructed from NAND gates by applying De Morgan's laws. Similarly, the NOT, AND, and OR gates can be constructed using NOR gates. This versatility allows for the creation of various truth tables and logic gates, demonstrating the importance of De Morgan's laws in digital electronics and computer science.
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De Morgan's first theorem
De Morgan's theorem, named after Augustus De Morgan (1806–1871), consists of two theorems. The first theorem can be represented by the equation:
> $\overline{A+B}=\overline{A}.\overline{B}$
This states that the complement of the product of two variables is corresponding to the sum of the complement of each variable. In other words, the NAND gate function is similar to the OR gate function with complemented inputs.
De Morgan's theorem can be applied to the negation of a disjunction or the negation of a conjunction in all or part of a formula. For example, if it is not the case that both P and Q are true, then at least one of P or Q is false, so P or Q is true.
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De Morgan's second theorem
De Morgan's theorem, a set of two laws, was formulated by the nineteenth-century British mathematician Augustus De Morgan. The laws are used to simplify large expressions in Boolean algebra and can be applied to the negation of a disjunction or the negation of a conjunction in all or part of a formula.
The first of De Morgan's laws can be illustrated by the following claim: "It is false that either of A or B is true". This means that neither A nor B is true, so it must follow that both A and B are not true. In other words, if either A or B were true, then the disjunction of A and B would be true, making its negation false.
The second theorem of De Morgan states that the inversion of the sum is the same as the product of the inversions. In other words, to simplify an expression using the De Morgan theorem, 'OR' is replaced with 'AND' and 'AND' is replaced with 'OR'. For example, if we consider the statement: "The numbers x and y are both odd", its negation can be expressed as: "The number x is even or the number y is even".
De Morgan's theorem holds true for two or more variables and can be applied to an expression under an inversion. The two theorems can be verified using a truth table.
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Using De Morgan's law in real-world scenarios
De Morgan's laws are an example of mathematical duality, which can be applied to the negation of a disjunction or the negation of a conjunction in all or part of a formula. The laws are named after Augustus De Morgan, who introduced a formal version of the laws to classical propositional logic.
De Morgan's laws are prevalent in logic and can be applied in real-world scenarios, such as in digital circuit design and formal logic. For example, in digital circuit design, De Morgan's laws can be used to manipulate the types of logic gates, helping to perform the same operation with fewer apparatus. In formal logic, De Morgan's laws are needed to find the conjunctive normal form and disjunctive normal form of a formula.
De Morgan's laws can also be applied to search queries to return the same set of documents. For instance, consider two searches: Search A, which looks for documents containing the word "cats" or "dogs", and Search B, which looks for documents that do not contain "cats" and also do not contain "dogs". Search A will return documents that mention either "cats" or "dogs", while Search B will return documents that do not mention either "cats" or "dogs". Applying De Morgan's laws, we can see that these two searches will return the same set of documents.
In Boolean algebra, De Morgan's first law states that the "complement of the union of two sets is equal to the intersection of the complement of the sets". This can be understood using Venn diagrams. De Morgan's laws can also be applied to logic gates, where they state that "both the logic gate circuits, i.e., NOT gate is added to the output of the OR gate, and NOT gate is added to the input of the AND gate, are equivalent".
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Frequently asked questions
De Morgan's Laws are rules of logic that describe how mathematical statements and concepts are related through their opposites. They are applicable in set theory, propositional logic, and computer engineering.
The first law states that the contradiction of "A and B" will be "not A or not B". The second law states that the contradiction of "A or B" will be "not A and not B".
De Morgan's Theorem is a fundamental principle in Boolean algebra that explains the relationship between the complement of the product of all terms and the sum of the complement of each term, and vice versa.
Consider the statement, "it is false that A and B are both true". For this claim to be true, either or both of A or B must be false. This can be written as, "Not (A and B) is the same as Not A or Not B".
