After identifying a mathematical object to study, one of the first things a mathematician will do is identify the properties or characteristics which that object has. The properties of mathematical objects form the basis for classification of those objects. It is also interesting to see the relationships between the properties that a an object may or may not possess. For example a mathematician may try to find out if possessing a certain property or set of properties will always mean that there is some other property that the object is certain to have (or not have).
Here are some examples of mathematical objects and properties that they may or may not have.
Counting numbers: Is the number even? odd? divisible by 9? prime? a square?
