Definition:Set

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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 Axiom of Abstraction.


However, there are problems with the Axiom of Abstraction. 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 universal set into two parts:

all the things that belong to $S$
all the things that do not belong to $S$.


Explicit Definition

A (finite) set can be defined by explicitly specifying all of its elements between the famous curly brackets, known as set braces: $\set {}$.


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

This is called explicit (set) definition.


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

$S = \set {a, \set a}$


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

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 $S = \set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$.

A more compact way of defining this set is:

$S = \set {1, 2, \ldots, 10}$

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 $10$.


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


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 (characteristic) property $P$ is either true or false (in Aristotelian logic) and so can be the subject of a propositional function $\map P x$.

Hence a set can be specified by means of such a propositional function:

$S = \set {x: \map P x}$

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 $\map P x$ is true.

We can express this symbolically as:

$\forall x: \paren {x \in S \iff \map P x}$

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


Warning

It is important to distinguish between an element, for example $a$, and a singleton containing it, that is, $\set a$.

That is $a$ and $\set a$ are not the same thing.

While it is true that:

$a \in \set a$

it is not true that:

$a = \set a$

neither is it true that:

$a \in a$


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.


Also known as

In the original translation by Jourdain of Georg Cantor's original work, this concept was called an aggregate or aggregation.

The term can be seen in subsequent works, but has now mostly been superseded by the term set.


Sometimes the terms class, family, system or collection are used.

In some contexts, the term space is used.

However, beware that these terms are usually used for more specific things than just as a direct synonym for set.

Hence on $\mathsf{Pr} \infty \mathsf{fWiki}$, these are not to be used as synonyms for set, being reserved specifically for the concepts to which they apply.


Point Set

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

In particular, the Cartesian plane and complex plane can each be seen referred to as a two-dimensional point set.


Examples

Set of Living People

Let $P$ denote the set of living people.

Then:

\(\ds \text {The person reading this web page}\) \(\in\) \(\ds P\)
\(\ds \text {Julius Caesar}\) \(\notin\) \(\ds P\)
\(\ds -4\) \(\notin\) \(\ds P\)


Positive Integers Less than 10

The (strictly) positive integers less than $10$ form a set:

$\set {1, 2, 3, 4, 5, 6, 7, 8, 9}$


Also see

  • Results about set theory can be found here.


Historical Note

The concept of a set first appears in Bernhard Bolzano's posthumous ($1851$) work Paradoxien des Unendlichen (The Paradoxes of the Infinite).

The first investigation into the concept in any depth was made by Georg Cantor in his two papers called Beiträge zur Begründung der transfiniten Mengenlehre ($1895$ and $1897$).

It was Georg Cantor who, in $1874$, defined a set thus:

By a set $M$ we understand any collection into a whole of definite and separate objects $m$ of our intuition or our thought.

Hence the definition of a set as:

a Many that allows itself to be thought of as a One.
-- Georg Cantor, A. Fraenkel and E. Zermelo, Gesammelte Abhandlungen (Berlin: Springer-Verlag, $1932$)

This definition was directly inspired by a problem raised by Bernhard Riemann in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe of $1854$, on the subject of Fourier series.


Internationalization

Set is translated:

In Catalan: conjunt
In Chinese: 集合
In French: ensemble
In German: Menge  (literally: aggregate)


Sources

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