Many examples of Noetherian topological spaces come from algebraic geometry, where for the Zariski topology an irreducible set has the intuitive property that any closed proper subset has smaller dimension.
42.
Therefore, the set of functions computable by LOOP-programs is a proper subset of computable functions ( and thus a subset of the computable by WHILE and GOTO program functions ).
43.
It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.
44.
It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.
45.
Every recurrent point is a nonwandering point, hence if f is a homeomorphism and X is invariant subset of the non-wandering set of f ( and may be a proper subset ).
46.
Thus, when only a proper subset of all the principal components are selected for regression, the PCR estimator so obtained is based on a hard form of orthogonal to the excluded directions.
47.
By showing there is a bijection between the set of natural numbers and one of its proper subsets, you have simply proved that the set of natural numbers is a Dedekind-infinite set.
48.
Set T is a " minimal complex " of X if and only if X depends on T and no proper subset S of T exists such that X depends on S.
49.
A simpler example is to consider the obvious bijection between the set of natural numbers and its proper subset of even numbers . talk ) 08 : 16, 13 June 2009 ( UTC)
50.
Note that the existence of a proper subset of a set that has the uniqueness property " cannot " in general be used as evidence that the superset is not a candidate key.
