Definition:Set

Definition
A set is intuitively defined as any aggregation of objects, called elements, which can be precisely defined in some way or other.

We can think of each set as a single entity in itself, and we can denote it (and usually do) by means of a single symbol.

That is, anything you care to think of can be a set. This concept is known as the comprehension principle.

However, there are problems with the comprehension principle. If we allow it to be used without any restrictions at all, paradoxes arise, the most famous example probably being Russell's Paradox.

Hence some sources define a set as a  'well-defined' collection of objects, leaving the concept of what constitutes well-definition to later in the exposition.

Defining a Set
The elements in a set $S$ are the things that define what $S$ is.

If $S$ is a set, and $a$ is one of the objects in it, we say that $a$ is an element (or member) of $S$, or that $a$ belongs to $S$, or $a$ is in $S$, and we write $a \in S$.

If $a$ is not one of the elements of $S$, then we can write $a \notin S$ and say $a$ is not in $S$.

Thus a set $S$ can be considered as dividing the universe into two parts:
 * all the things that belong to $S$
 * all the things that do not belong to $S$.

Explicit Definition
If there are many elements in a set, then it becomes tedious and impractical to list them all in one big long explicit definition. Fortunately, however, there are other techniques for listing sets.

Implicit Definition
Explicit and implicit definition are collectively referred to as roster notation.