On Property

The word property is not easy to define precisely.
According to Merriam Webster: 'property' can be interpreted as:

(a) A quality or trait belonging and especially peculiar to an individual or thing;
(b) An effect that an object has on another object or on the senses;
(c) An attribute common to all members of a class.

More simple, according to Google Dictionary:
(a) A thing or things that are owned by somebody; a possession or possessions;
(b) A quality or characteristic that something has.

Mathematically, however, we shall not hesitate to use it in the usual (informal) fashion.
If P denotes a property that is meaningful for a collection of elements, then we agree to write {x : P(x)} for the set of all elements x for which the property P holds. We usually read this as "the set of all x such that P(x)". It is often worthwhile to specify which elements we are testing for the property P. Hence we shall often write:

{x  S : P(x)} for the subset of S for which the property P holds.