Alicia Boole Stott

In computer science, a tree is a widely used abstract data type that simulates a hierarchical tree structure

Model theory recognizes and is intimately concerned with a duality: it examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language

In fact, the Product comonad is just the dual of the Writer monad and effectively the same as the Reader monad (both discussed below)

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

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

Dual (mathematics), a notion of paired concepts that mirror one another

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.

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".

This is an abstract form of De Morgan's laws, or of duality applied to lattices.

A plane graph is said to be self-dual if it is isomorphic to its dual graph.

In mathematical contexts, duality has numerous meanings[1] although it is "a very pervasive and important concept in (modern) mathematics"[2] and "an important general theme that has manifestations in almost every area of mathematics".[3]

the phrase up to is used to convey the idea that some objects in the same class — while distinct — may be considered to be equivalent under some condition or transformation

"a and b are equivalent up to X" means that a and b are equivalent, if criterion X, such as rotation or permutation, is ignored

If solutions that differ only by the symmetry operations of rotation and reflection of the board are counted as one, the puzzle has 12 solutions. These are called fundamental solutions; representatives of each are shown below

In Hardy's words, "Exposition, criticism, appreciation, is work for second-rate minds. [...] It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done."

A rooted binary tree is full if every vertex has either two children or no children.

A computable Dedekind cut is a computable function which when provided with a rational number as input returns or ,