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.

However, there are problems with this concept. If we allow this definition to be used without any restrictions at all, paradoxes arise, for example Russell's paradox.

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 $$S$$.

Sometimes elements are referred to as points, and sets referred to as point sets.

One way of defining a set is by 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.

Notes