10 Matching Annotations
1. Jul 2020
2. en.wikipedia.org en.wikipedia.org
1. In logic, functions or relations A and B are considered dual if A(¬x) = ¬B(x), where ¬ is logical negation. The basic duality of this type is the duality of the ∃ and ∀ quantifiers in classical logic. These are dual because ∃x.¬P(x) and ¬∀x.P(x) are equivalent for all predicates P in classical logic
2. the ∧ and ∨ operators are dual in this sense, because (¬x ∧ ¬y) and ¬(x ∨ y) are equivalent. This means that for every theorem of classical logic there is an equivalent dual theorem. De Morgan's laws are examples

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3. May 2020
4. en.wikipedia.org en.wikipedia.org
1. Related concepts in other fields are: In natural language, the coordinating conjunction "and". In programming languages, the short-circuit and control structure. In set theory, intersection. In predicate logic, universal quantification.

Strictly speaking, are these examples of dualities (https://en.wikipedia.org/wiki/Duality_(mathematics))? Or can I only, at strongest, say they are analogous (a looser coonection)?

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5. en.wikipedia.org en.wikipedia.org

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6. en.wikipedia.org en.wikipedia.org
1. Mathematically speaking, necessity and sufficiency are dual to one another. For any statements S and N, the assertion that "N is necessary for S" is equivalent to the assertion that "S is sufficient for N".

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7. en.wikipedia.org en.wikipedia.org

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8. en.wikipedia.org en.wikipedia.org
1. This is an abstract form of De Morgan's laws, or of duality applied to lattices.

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9. en.wikipedia.org en.wikipedia.org
1. A plane graph is said to be self-dual if it is isomorphic to its dual graph.

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10. en.wikipedia.org en.wikipedia.org
1. In mathematical contexts, duality has numerous meanings although it is "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".