De Morgan's laws - Wikipedia, the free encyclopedia.pdf

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De Morgan's laws

From Wikipedia, the free encyclopedia

In logic, De Morgan's laws (orDe Morgan's theorem) are rules in formal logic relating pairs of

dual logical operators in a systematic manner expressed in terms of negation. The relationship so

induced is called De Morgan duality.

The rule states that each of the following claims is logically equivalent to the one next to it and may

be legally transformed from one to the other in either direction:

Contents1 Logical applications

1.1 Negation of a disjunction

1.2 Negation of a conjunction

2 History and formulations

3 Further explanation

4 See also

5 External links

Logical applications

DeMorgan's theorem may be applied to the negation of a disjunction or the negation of a conjunction

in all or part of a formula.

Negation of a disjunction

In the case of its application to a disjunction, consider the following claim: it is false that either A or B

is true, which is written as:

In that it has been established that neitherA nor B is true, then it must follow that A is not true and B

is not true; If either A or B were true, then the disjunction of A and B would be true, making its

negation false.

Working in the opposite direction with the same type of problem, consider this claim:

This claim asserts that A is false and B is false (or "not A" and "not B" are true). Knowing this, adisjunction of A and B would be false, also. However the negation of said disjunction would yield a

true result that is logically equivalent to the original claim. Presented in English, this would follow the

logic that "Since two things are false, it's also false that either of them are true."

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Negation of a conjunction

The application of DeMorgan's theorem to a conjunction is very similar to its application to a

disjunction both in form and rationale. Consider the following claim: It is false that A and B are both

true, which is written as:

In order for this claim to be true, either or both of A or B must be false, in that if they both were true,

then the conjunction of A and B would be true, making its negation false. So, the original claim may

be translated as "Either A is false or B is false", or "Either not A is true or not B is true".

Presented in English, this would follow the logic that "Since it is false that two things together are

true, at least one of them must be false."

History and formulations

Augustus De Morgan originally observed that in classical propositional logic the following

relationships hold:

not (P and Q) = (not P) or (not Q)

not (P or Q) = (not P) and (not Q)

De Morgan's observation influenced the algebraisation of logic undertaken by George Boole, which

cemented De Morgan's claim to the find, although a similar observation was made by Aristotle and

was known to Greek and Medieval logicians (cf. Bocheski'sHistory of Formal Logic).

Further explanation

In formal logic the laws are usually written

and in set theory

In extensions of classical propositional logic, the duality still holds (that is, to any logical operator we

can always find its dual), since in the presence of the identities governing negation, one may always

introduce an operator that is the De Morgan dual of another. This leads to an important property of

logics based on classical logic, namely the existence of negation normal forms: any formula is

equivalent to another formula where negations only occur applied to the non-logical atoms of the

formula. The existence of negation normal forms drives many applications, for example in digital

circuit design, where it is used to manipulate the types of logic gates, and in formal logic, where it is aprerequisite for finding the conjunctive normal form and disjunctive normal form of a formula.

Computer programmers use them to change a complicated statement like IF ... AND (... OR ...)

THEN ... into its opposite. They are also often useful in computations in elementary probability

theory.



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Let us define the dual of any propositional operator P(p, q, ...) depending on elementary propositions

, q, ... to be the operator Pddefined by

This idea can be generalised to quantifiers, so for example the universal quantifier and existential

quantifier are duals:

To relate these quantifier dualities to the De Morgan laws, set up a model with some small number of

elements in its domainD, such as

D = {a, b, c}.

Then

and

But, using De Morgan's laws,

and

verifying the quantifier dualities in the model.

Then, the quantifier dualities can be extended further to modal logic, relating the box ("necessarily")

and diamond ("possibly") operators:

In its application to the alethic modalities of possibility and necessity, Aristotle observed this case, andin the case of normal modal logic, the relationship of these modal operators to the quantification can

be understood by setting up models using Kripke semantics.

In C, Java, and other related programming languages, De Morgan's laws can be written as:

!(p && q) == !p || !q

!(p || q) == !p && !q

These equations always return a value of true, regardless of the values of p and q.

In electrical engineering contexts, the negation operator can be written as an overline above the terms

to be negated. Thus, electrical engineering students are often taught to remember DeMorgan's laws

using the mnemonic "break the line, change the sign".



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See also

List of Boolean algebra topics

External links

Eric W. Weisstein, de Morgan's Laws (http://mathworld.wolfram.com/deMorgansLaws.html)at

MathWorld.

de Morgan's laws (http://planetmath.org/encyclopedia/DeMorgansLaws.html)at PlanetMath.

Retrieved from "http://en.wikipedia.org/wiki/De_Morgan%27s_laws"

Categories: Boolean algebra | Logic | Duality theories | Rules of inference

This page was last modified 20:44, 22 August 2007.

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