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3 is the only integer which is the sum of the preceding positive integers (1+2=3) and the only number which is the sum of the factorials of the preceding positive integers ...

Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems. The most important of these ...

An affine geometry is a geometry in which properties are preserved by parallel projection from one plane to another. In an affine geometry, the third and fourth of Euclid's ...

The approximation problem is a well known problem of functional analysis (Grothendieck 1955). It asks to determine whether every compact operator T from a Banach space X to a ...

Backgammon is a board game played by two players. The backgammon board is divided into four sections of six triangular regions on which pieces may be placed. Brown and white ...

Every Boolean algebra is isomorphic to the Boolean algebra of sets. The theorem is equivalent to the maximal ideal theorem, which can be proved without using the axiom of ...

The term Borel hierarchy is used to describe a collection of subsets of R defined inductively as follows: Level one consists of all open and closed subsets of R, and upon ...

In the theory of transfinite ordinal numbers, 1. Every well ordered set has a unique ordinal number, 2. Every segment of ordinals (i.e., any set of ordinals arranged in ...

A module having dual properties with respect to a free module, as enumerated below. 1. Every free module is projective; every cofree module is injective. 2. For every module ...

A product of ANDs, denoted ^ _(k=1)^nA_k. The conjunctions of a Boolean algebra A of subsets of cardinality p are the 2^p functions A_lambda= union _(i in lambda)A_i, where ...