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A function f mapping a set X->X/R (X modulo R), where R is an equivalence relation in X, is called a canonical map.

Each point in the convex hull of a set S in R^n is in the convex combination of n+1 or fewer points of S.

A set X is a called a "convex cone" if for any x,y in X and any scalars a>=0 and b>=0, ax+by in X.

Suppose <= is a partial ordering on a nonempty set A. Then the elements a,b in A are said to be comparable provided a<=b or b<=a. Because two elements in a partially ordered ...

Let S be a collection of subsets of a set X and let mu:S->[0,infty] be a set function. The function mu is called a premeasure provided that mu is finitely additive, countably ...

A set is said to be bounded from above if it has an upper bound. Consider the real numbers with their usual order. Then for any set M subset= R, the supremum supM exists (in ...

A set is said to be bounded from below if it has a lower bound. Consider the real numbers with their usual order. Then for any set M subset= R, the infimum infM exists (in R) ...

The set P^2 is the set of all equivalence classes [a,b,c] of ordered triples (a,b,c) in C^3\(0,0,0) under the equivalence relation (a,b,c)∼(a^',b^',c^') if ...

The tangent plane to a surface at a point p is the tangent space at p (after translating to the origin). The elements of the tangent space are called tangent vectors, and ...

Given a set A, let N(A) be the set of neighbors of A. Then the bipartite graph G with bipartitions X and Y has a perfect matching iff |N(A)|>=|A| for all subsets A of X.