Property

This term is 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 haves and the other labelled the have-nots
• 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