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De Morgan's laws - Wikipedia

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De Morgan's laws - Wikipedia

In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference.

They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression‎ of conjunctions and disjunctions purely in terms of each other via negation.



The rules can be expressed in English as:

The negation of "A and B" is the same as "not A or not B".

The negation of "A or B" is the same as "not A and not B".

or

The complement of the union of two sets is the same as the intersection of their complements

The complement of the intersection of two sets is the same as the union of their complements

or

not (A or B) = (not A) and (not B)

not (A and B) = (not A) or (not B)

where "A or B" is an "inclusive or" meaning at least one of A or B rather than an "exclusive or" that means exactly one of A or B.




Another form of De Morgan's law is the following as seen below.
A−(B∪C)=(A−B)∩(A−C),
A−(B∩C)=(A−B)∪(A−C).

Applications of the rules include simplification of logical expressions in computer programs and digital circuit designs. De Morgan's laws are an example of a more general concept of mathematical duality.


● 명제논리
Propositional Logic


Propositional logic, also known as sentential logic and statement logic, is the branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences, as well as the logical relationships and properties that are derived from these methods of combining or altering statements. In propositional logic, the simplest statements are considered as indivisible units, and hence, propositional logic does not study those logical properties and relations that depend upon parts of statements that are not themselves statements on their own, such as the subject and predicate of a statement. The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic, which studies logical operators and connectives that are used to produce complex statements whose truth-value depends entirely on the truth-values of the simpler statements making them up, and in which it is assumed that every statement is either true or false and not both. However, there are other forms of propositional logic in which other truth-values are considered, or in which there is consideration of connectives that are used to produce statements whose truth-values depend not simply on the truth-values of the parts, but additional things such as their necessity, possibility or relatedness to one another.



● 불리언(Boolean)

불리언(Boolean)은 참과 거짓을 나타내는 데이터 타입으로, 논리 자료형이라고도 합니다.

컴퓨터 과학에서 사용되며, bool이라고도 불립니다.


불리언의 특징

참을 의미하는 true와 거짓을 의미하는 false 두 가지의 값을 가지고 있습니다.

○ 주로 참은 1, 거짓은 0에 대응하나 언어마다 차이가 있습니다.
○ 숫자를 쓰지 않고 참과 거짓을 나타내는 영단어 true와 false를 사용합니다.

불리언은 프로그래밍 언어에서 사용되며, 다음과 같은 언어에서 사용됩니다. 자바(Java), Visual Basic, C#, JavaScript, 아두이노.