Definition:Set

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.

Sometimes the terms class, family or collection are used. In some contexts, the term space is used. However, beware that these terms can be used for more specific things than just as a synonym for "set".

Cantor defined a set as being:
 * "a Many that allows itself to be thought of as a One."

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.

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 $$S$$.

Explicit Definition
One way of defining a set is by explicitly specifying all of its elements between the famous curly brackets, or "set braces": $$\left\{{}\right\}$$. For example, the following define sets:

$$S = \left\{{\textrm {Tom,~ Dick,~ Harry}}\right\}$$

$$T = \left\{{1, 2, 3, 4}\right\}$$

$$V = \left\{{\textrm {red,~ orange,~ yellow,~ green,~ blue,~ indigo,~ violet}}\right\}$$

When a set is defined like this, note that all and only the elements in it are listed.

This is called explicit definition.

It is possible for a set to contain other sets. For example:

$$S = \left\{ {a, \left\{ {a}\right\}}\right\}$$

Note here that $$a$$ and $$\left\{{a}\right\}$$ are not the same thing.

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

If the elements in a set have an obvious pattern to them, we can define the set implicitly by using an ellipsis ($$\ldots$$).

For example, suppose $$S = \left\{ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \right\}$$. A more compact way of defining this set is: $$S = \left\{ {1, 2, \ldots, 10} \right\}$$. With this notation we are asked to suppose that the numbers count up uniformly, and we can read this definition as "$$S$$ is the set containing $$1$$, $$2$$, and so on, up to $$10$$."

See how this notation is used: we have a comma before the ellipsis and one after it. It is a very good idea to be careful with this.

The point needs to be made: "how obvious is obvious?" If there is any doubt as to the precise interpretation of an ellipsis, either the rule should be explicitly specified, or it should be left out.

Definition by Predicate
An object can be specified by means of a predicate, that is, in terms of a property (or properties) that it possesses.

Whether an object $$x$$ possesses a particular property $$P$$ is either true or false (in Aristotelian logic) and so can be the subject of the propositional function $$P \left({x}\right)$$.

Hence a set can be specified by means of such a propositional function, e.g.:
 * $$S = \left\{{x: P \left({x}\right)}\right\}$$

which means: "$$S$$ is the set of all objects which have the property $$P$$", or, more formally: "$$S$$ is the set of all $$x$$ such that $$P \left({x}\right)$$ is true."

In this context, we see that the symbol $$:$$ is interpreted as "such that".

An alternative notation for this is $$S = \left\{{x| P \left({x}\right)}\right\}$$, but it can be argued that the use of $$|$$ for "such that" can cause ambiguity and confusion, as $$|$$ has several other meanings in mathematics.

Uniqueness of Elements
A set is uniquely determined by its elements. This means that the only thing that defines what a set is is what it contains. So, how you choose to list or define the contents makes no difference to what the contents actually are.

Note these points:

For a given set, an object is either in the set or not in the set. So, if an element is in a set, then it is in the set only once, however many times it may appear in the definition of the set. Thus, the set $$\left\{ {1, 2, 2, 3, 3, 4}\right\}$$ is the same set as $$\left\{ {1, 2, 3, 4}\right\}$$. $$2$$ and $$3$$ are in the set, and listing them twice makes no difference to the set contents. Like the membership of a club, if you're in, you're in - however many membership cards you have to prove it.

It makes no difference what order the elements are specified. This means that the sets $$S = \left\{ {1, 2, 3, 4}\right\}$$ and $$T = \left\{ {3, 4, 2, 1}\right\}$$ are the same set.

Two sets which have exactly the same elements are the same, whatever the sets are called. So, to take the club membership analogy, if two clubs had exactly the same members, the clubs would be considered as the same club, although they may be given different names. This follows from the definition of "equals" given above.

Note that there are mathematical constructs which do take into account both (or either of) the order in which the elements appear, and the number of times they appear, but these are not "sets" as such.

Point Set
A set whose elements are all (geometric) points is often called a point set.

Historical Note
Although the concept of a set as currently understood originates mainly with Georg Cantor, it first appears in Bolzano's posthumous (1851) work  Paradoxien des Unendlichen (The Paradoxes of the Infinite).