In mathematics, a complete set of invariants for a classification problem is a collection of maps (where
is the collection of objects being classified, up to some equivalence relation
In words, such that two objects are equivalent if and only if all invariants are equal.
[1] 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 also 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).