space supplemental angles
Any abstract set with a structure defined on it, such as an order relation, metric, etc.
Cf. Euclidean space, Hilbert space, metric space, topological space.
A closed surface, all points of which are equidistant from a given point, called the center.
In 3-dimensional Euclidean space, the equation of a sphere of radius r and center (h, j, k) is
The term sphere may also refer to the solid bounded by this surface, and the interior is then called the open sphere of radius r.
More generally, a sphere may be defined as the set of points in n-dimensional space (or any metric space) equidistant from a given point. The unit sphere in n-dimensional space is typically denoted S n - 1. Thus, the unit sphere in ordinary 3-space is denoted S2, and the unit circle in the plane is denoted S1.
A regular polygon having four equal sides and four right angles.
A matrix that has the same number of rows as columns.
If a is an ordinal, a set S in a is called stationary if S has non-empty intersection with every closed unbounded subset of a.
If X is a compact space and C(X) denotes the space of all continuous functions on X, and A is an algebra of functions in C(X) which separates the points of C(X) and which contains a constant function f not identically zero, then A is dense in C(X).
A mathematical problem presented as a real-world situation. See the article for problem solving techniques.
In a topological space, a subbase is a collection of sets, the collection of all finite intersections of which constitutes a base.
A subset of a field which is also a field, with the operations defined by restriction.
A subset of a group which is also a group. Every subgroup induces a partition of the original group, and if this partition is itself a group under the inherited group operation, the subgroup is called a normal subgroup.
We say that Y is a sublattice of a lattice X if Y is a subset of X and if a b, a b are in Y whenever a, b are in Y.
A subset of a ring which is also a ring, with the operations defined by restriction.
A set A is a subset of a set B if every element of A is also an element of B. If in addition B is a subset of A, then A = B, but if not then A may be said to be a proper subset of B.
Intuitively, the assertion that if one “plugs” a value into a formal polynomial, everything will work out as expected.
Formally, the theorem that, given two commutative rings R and R', a homomorphism f from R to R', and an element r of R', there is a unique homomorphism F from the polynomial ring R[x] over R to R', with the properties
For any polynomial p, Fr(p) is written p(r), and corresponds to “evaluating p on r.”
- F(a) = f(a) for all a in R,
- F(x) = r.
To subtract a number m from a number n is to calculate the difference of m and n. If m is less than n we take the positive difference, otherwise we take the negative of the difference. This is tantamount to adding the negative of m to n.
In a structure with an order relation defined upon it, the successor of an element a is the least element greater than a, if such exists.
Given a set A, the sumset of A, denoted by
is the set containing all of the elements of the elements of A, that is, it is the union of the elements of A.
An axiom of set theory which states that if A is any set, then the sumset of A is also a set.
Abbreviation of supremum.
A set A is a superset of a set B if every element of B is an element of A.
Two angles are supplemental if they add up to 180 degrees (p radians).
Cf. complementary angles.