# Property

From Groupprops

This term is related to: property theory

View other terms related to property theory | View facts related to property theory

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## Definition

A **property** over a collection of objects, is any of the following equivalent things:

- A map from the collection of objects, to the two-element set (true, false). Those elements which map to true are said to
*have*the property, and those which map to false are said to*not have*the property - A partition of the collection of objects into two subcollections, with one subcollection labelled the
*have*s and the other labelled the*have-not*s - A choice of a subcollection of the given collection

The collection of objects is termed the context space for the property. Typically, when we talk of properties over a collection of mathematical objects, we want the property to be over *isomorphism classes* of objects; two isomorphic objects should either both have the property or both not have the property.

## Related notions

### Property space

`Further information: property space`