Complete Set Of Invariants
In mathematics, a complete set of invariants for a classification problem is a collection of maps
(where X is the collection of objects being classified, up to some equivalence relation, and the are some sets), such that ∼ if and only if for all i. In words, such that two objects are equivalent if and only if all invariants are equal.
Symbolically, a complete set of invariants is a collection of maps such that
is injective.
As invariants are, by definition, equal on equivalent objects, equality of invariants is a necessary condition for equivalence; a complete set of invariants is a set such that equality of these is sufficient for equivalence. In the context of a group action, this may be stated as: invariants are functions of coinvariants (equivalence classes, orbits), and a complete set of invariants characterizes the coinvariants (is a set of defining equations for the coinvariants).
Read more about Complete Set Of Invariants: Examples, Realizability of Invariants
Famous quotes containing the words complete and/or set:
“It is ... pathetic to observe the complete lack of imagination on the part of certain employers and men and women of the upper-income levels, equally devoid of experience, equally glib with their criticism ... directed against workers, labor leaders, and other villains and personal devils who are the objects of their dart-throwing. Who doesnt know the wealthy woman who fulminates against the idle workers who just wont get out and hunt jobs?”
—Mary Barnett Gilson (1877?)
“Ill give my jewels for a set of beads,
My gorgeous palace for a hermitage,
...
And my large kingdom for a little grave,
A little, little grave, an obscure grave.”
—William Shakespeare (15641616)