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1 Introduction
A set is a collection, group, or conglomerate^{1}^{1}However, not every collection has to be a set (in fact, all collections can’t be sets: there is no set of all sets or of all ordinals for example). See proper class for more details..
Sets can be of “real” objects or mathematical objects, but the sets themselves are purely conceptual. This is an important point to note: the set of all cows (for example) does not physically exist, even though the cows do. The set is a “gathering” of the cows into one conceptual unit that is not part of physical reality. This makes it easy to see why we can have sets with an infinite number of elements; even though we may not be able to point out infinitely many objects in the real world, we can construct conceptual sets which an infinite number of elements (see the examples below).
The symbol $\in$ denotes set membership. For example, $s\in S$ would be read “$s$ is an element of $S$”.
We write $A\subset B$ if for all $x\epsilon A$ we have $x\epsilon B$ and we then say $B$ contains $A$. We sometimes write “$S$ contains $s$” when $S$ contains the set whose only element is $s$.
Mathematics is thus built upon sets of purely conceptual, or mathematical, objects. Sets are usually denoted by uppercase roman letters (such as $S$). Sets can be defined by listing the members, as in
$S:=\{a,b,c,d\}.$ 
Or, a set can be defined from a predicate (called “set builder notation”). This type of statement defining a set is of the form
$S:=\{x\in X:P(x)\},$ 
where $S$ is the symbol denoting the set, $x$ is the variable we are introducing to represent a generic element of the set (note that, by the so called axiom of comprehension (or axiom of subsets), $x$ must be a member of some set which has already been defined. This is necessary in order to avoid Russell’s paradox^{2}^{2}One needs to be careful when defining a set by a predicate only, since (for example) “$x$ is not in $x$” is a perfectly good predicate. Either one needs to restrict the kind of predicate, or, more commonly, one needs to define only subsets by predicates. So while one cannot do $\{x\colon x\not\in x\}$, if one already has a set $U$, one can do $\{x:x\in U,x\not\in x\}$..) and $P(x)$ is some property which must be true for any element $x$ of the set (that is, $x\in S$ is equivalent to $x\in X$ and $P(x)$ holds.) Sometimes, we write a set definition as $S:=\{f(x)\in X:P(x)\}$, where $f(x)$ is a transformation of that variable. In this case, we can simply replace the set $X$ by $Y$, where $Y:=\{y\in X:\exists x\in X,y=f(x)\}$ in order to define the set as above.
Sets are, in fact, completely specified by their elements. If two sets have the same elements, they are equal. This is called the axiom of extensionality, and it is one of the most important characteristics of sets that distinguishes them from predicates or properties.
Some examples of sets are:

The standard number sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$.

The set of all even integers: $\{x\in\mathbb{Z}:2\!\mid\!x\}.$

The set of all prime numbers (sometimes denoted $\mathbb{P}$): $\{p\in\mathbb{N}:p>1,\forall x\in\mathbb{N}\;\;x\!\mid\!p\Rightarrow x\in\{1,p\}\}$, where $\Rightarrow$ denotes implies and $\mid$ denotes divides.

The set of all real functions of one real parameter (sometimes denoted by $\mathbb{R}^{\mathbb{R}}$): $\{f(x)\in\mathbb{R}:x\in\mathbb{R}\}$ or, more formally, $\{f\subset\mathbb{R}^{2}:\left[\forall x\in\mathbb{R},\exists y\in\mathbb{R},(% x,y)\in f\right]\wedge\left[(x,y),(x,y^{{\prime}})\in f\Rightarrow y=y^{{% \prime}}\right]\}$.

The unit circle $\mathbb{S}^{1}$: $\{z\in\mathbb{C}:z=1\}$, where $z$ is the modulus of $z$.
The astute reader may have noticed that all of our examples of sets utilize sets, which does not suffice for rigorous definition. We can be more rigorous if we postulate only the empty set, and define a set in general as anything which one can construct from the empty set and the ZFC axioms. The nonnegative integers, for instance, are defined by $0:=\varnothing$ and the successor of $x$, $s(x)=x\cup\{x\}.$ A nonnegative integer is thus the set of all its predecessors (for example, we have $3=\{0,1,2\}=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}$)^{3}^{3}Note however that the existence of the set of nonnegative integers needs an additional axiom beside those which are required to define its members: the axiom of infinity..
All objects in modern mathematics are constructed via sets. An important point to be made about this is that the construction of the object is less important than the way it will behave. As an example, we usually define an ordered pair $(x,y)$ as the set $\{x,\{x,y\}\}$: what matters here is that, for two ordered pairs $(x,y)$ and $(x^{{\prime}},y^{{\prime}})$, we have $(x,y)=(x^{{\prime}},y^{{\prime}})$ if and only if $x=x^{{\prime}}$ and $y=y^{{\prime}}$, and this is true with the given definition, as one can easily see. We could, however, also have taken $\{x,\{x,\{y\}\}\}$ as the definition of $(x,y)$, in which case the needed property also holds and we have a valid definition (we chose the first only because it is simpler).
2 Set Notions
An important set notion is cardinality. Cardinality is roughly the same as the intuitive notion of “size” or number of elements. While this intuitive definition works well for finite sets, intuition breaks down for sets with an infinite number of elements. The cardinality of a set $S$ is denoted $S$ (sometimes $\#S$ or $\card S$) and we say that sets $A$ and $B$ have the same cardinality if and only if there is a bijection from one to the other. For more detail, see the cardinality entry.
Another important set concept is that of subsets. A subset $B$ of a set $A$ is any set which contains only elements that appear in $A$. Subsets are denoted with the $\subseteq$ symbol, i.e. $B\subseteq A$ (in which case $A$ is called a superset of B). Also useful is the notion of a proper subset, denoted $B\subsetneq A$ (or sometimes, $B\subset A$)^{4}^{4}Beware — some authors use $\subset$ to mean proper subset, while most use it to mean subset with equality (the same as $\subseteq$), which can make the $B\subset A$ notation ambiguous., which adds the restriction that $B$ must also not be equal to $A$. The set of all subsets of a set $S$ is called the power set of $S$, denoted $\mathcal{P}(S)$ (the existence of this set is also axiomatic: it is guaranteed by the axiom of the power set). Note that $B$ does not need to have a lower cardinality than $A$ to be a proper subset, i.e., $\{1,2,3,\ldots\}$ is a proper subset of $\{0,1,2,3,\ldots\}$, but both have the same cardinality, $\aleph_{0}$ (In fact, a set is infinite if and only if it has the same cardinality as some proper subset).
3 Set Operations
There are a number of standard (common) operations which are used to manipulate sets, producing new sets from combinations of existing sets (sometimes with entirely different types of elements). These standard operations are:
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fix linking policy by Mathprof ✓
spelling by Mathprof ✓
$\in$ should be $\subset$ by silverfish ✓
change hyperlink by kfgauss70 ✓
Comments
\subset vs. \subseteq
Hi
I think this entry defines \subset and \subseteq in a nonstandard way.
Usually (for instance in Rudin, Real and complex analysis),
A \subset B means that x\in A implies x\in B.
In particular, there is no restriction that A\neq B, and \subseteq in not even
defined. Other standard books seem to give the same definition.
I think that \subseteq is just a "synonym" for \subset which emphasizes that
equality is also possible. Using both, however, would be confusing since
then \subset would imply that the a subset is proper, which would be a
nonstandard interpretation.
I would suggest to only define \subset, and in a remarksection mention that
the symbol \subseteq is also sometimes used to emphasize that equality
is also possible in \subset. If \subset and \subseteq are used systematically
as in given definition, there is not problem, but since this is not the standard
definition, this seem to be a bit stretched. This could be a cultural thing
 I don't know. Therefore this is posted as a message and not a correction.
matte
Re: \subset vs. \subseteq
Yes, common usage is \subset is a synonym of \subseteq. But I feel that's more like a sloppy usage that went away}}I think of them as in lessthan and lessthanorequal signs
when this logic struck me I could never use \subset as before
So now (in my personal writings) when I write A \subset B I think of
A (less than) B  proper inclusion
(which is consisten with the order induced on Sets byt eh inclusion, so both \subset and (less than) are statements of order
and subseteq allows equality as well (just like the order relatio \leq) allows it
it's the logical consistent way to me
but that's me, and I know I can't impose such conventions, abnd so, whereas sometimes it slips on some entry, I stick to the synonym usage
(but again, I feel it's proper to make the idnstinctino and wish it would be more widespread)
f
G > H G
p \ /_  ~ f(G)
\ / f ker f
G/ker f
Re: \subset vs. \subseteq
Many books use \subseteq all the time, not just when they want to emphasize that equality is possible. So I think your suggested wording is misleading. But I agree that the article should mention the other usage, since it is quite common.
Like drini, I use \subset to mean "is a proper subset of" in my own notes (for the same reason that he does). Articles on PlanetMath should probably avoid using \subset altogether, because of the ambiguity.
Re: \subset vs. \subseteq
The problem with \subset is that it is potentially confusing, because different authors do use it in different ways. (Imagine if someone started using < when they meant <=  this would cause no end of trouble.)
In the context of a book, or other sustained piece of writing, it doesn't matter, because the author can indicate what the notation means in an introductory section.
Personally, I've always avoided the issue by using \subseteq and \subsetneq, avoiding \subset altogether. (Actually, I usually use \varsubsetneq  there are 4 different versions of this symbol.)
Re: \subset vs. \subseteq
> Many books use \subseteq all the time, not just when they
> want to emphasize that equality is possible.
Could you give some examples of such books? So far, I have only found
it used in the undergraduate book Schramm: Introduction to
Real analysis.
Mathworld defines A\subseteq B to mean x\in A => x\in B, and
A\subset B to mean that A is a proper subset of B. See
http://mathworld.wolfram.com/Subset.html
Matte
Re: \subset vs. \subseteq
> > Many books use \subseteq all the time, not just when they
> > want to emphasize that equality is possible.
>
> Could you give some examples of such books? So far, I have
> only found
> it used in the undergraduate book Schramm: Introduction to
> Real analysis.
A brief perusal of my bookshelf reveals:
Hodel  An Introduction to Mathematical Logic: Uses only \subseteq
Steen and Seebach  Counterexamples in Topology: Uses both \subseteq and \subset; the first seems to be used when emphasizing the analogy with \leq
Halmos  Naive Set Theory: Uses only \subset
Pedersen  Analysis Now: Uses only \subset
Royden  Real Anlalysis: Uses only \subset
Mumford  Algebraic Geometry I, Complex Projective Varieties: Uses both \subset and \subseteq; the latter used to emphasize the possibility of equality.
Lang  Algebra: Uses only \subset; explicitly says it includes equality
Pontryagin  Foundations of Combinatorial Topology: Uses only \subset; explicitly says it includes equality
Spivak  Calculus: Avoids the notation entirely
Serre  A Course in Arithmetic: Avoids the notation entirely
So, to summarize:
 Ten books were surveyed.
 Zero used the convention described in this entry.
 Seven use a convention inconsistent with this definition.
 Five do not say so.
 One uses \subseteq in all cases.
 Four use \subset in all cases.
 Two use \subseteq for emphasis, drastically misleading someone who believes this entry.
 Two avoid the notation, preferring words to symbols.
So, the definition in this entry is not standard. Nor do books that use another convention always define it. A few use a convention that seems to follow it but will trick readers who believe this. Conversely, while I have seen the convention that \subset is strict, it is always explicitly defined.
Finally, there is a perfectly adequate symbol, \subsetneq, to use when strict containment is meant. In printed material of the sort contained in PlanetMath (where entries are supposed to stand independently and where brevity is of no value), I can see no reason to ever use \subset.
Re: \subset vs. \subseteq
> > Many books use \subseteq all the time, not just when
> > they want to emphasize that equality is possible.
>
> Could you give some examples of such books?
"A Course in Universal Algebra" by Stanley Burris and H. P. Sankappanavar is one example. This is available online in PDF format: http://www.thoralf.uwaterloo.ca/htdocs/ualg.html
A couple more I have to hand are "Algebra in the StoneCech Compactification" by Neil Hindman and Dona Strauss, and "An Introduction to Nonassociative Algebras" by Richard D. Schafer.
Re: \subset vs. \subseteq
subsetneq is not as widely used and it should be explained, and if so, why can't we explain subset vs subseteq?
Although the common sense tell me that just a few notes on the entry clearing several usages should suffice
(and authors using subset where they mean subseteq could perhaps mention it whenb it really matters (that is when the conclusion changes, orwhen theorem hypotesis behave different, etc)
f
G > H G
p \ /_  ~ f(G)
\ / f ker f
G/ker f
Re: \subset vs. \subseteq
At PM, I think we should only have one official notation in use. At least
for this issue since the different notation conventions are not logically
equivalent. In other words, without a fixed notation, it will be impossible
to interpret what, say, A\subset B means.
As an encyclopÃ¦dia, whatever notation is chosen should at least
in some way reflect what is "standard" use in general. However, as
there are many conventions for the use of \subset, \subseteq,
\subsetneq, \not\subset, all these should be discussed in this entry,
but only after the "official" definition is given.
Matte
Re: \subset vs. \subseteq
> subsetneq is not as widely used and it should be explained,
> and if so, why can't we explain subset vs subseteq?
You are right that it is rare (although not as rare as using \subset to mean the same thing) and should be explained in some entry somewhere (which it is). But it does not need to be explained every time it is used: almost anyone seeing it will know exactly what it means even if they have never seen it before. And it is never used to mean anything else, so users will not be confused by it.
Similarly, using \subseteq instead of \subset has no danger of confusing anyone: if they've seen either usage, they will know exactly what is meant, with no further explanation.
Using \subset, there is a possibility of confusion; the reader has to confirm that the author is using it in the standard way. (Most readers will assume this.)
I don't think there's any need to go pester every author who ever uses \subset to mean \subseteq unless a reader is genuinely confused by it; authors who use \subset to mean \subsetneq should be pestered to change it or at least say so.
Re: \subset vs. \subseteq
that's why I said "when it really matters" (on being specific about subset vs subseteq). I (personally) don't have problems with people using \subset when they mean \subseteq since , for most of the times, it's irrelevant.
I try however to use \subseteq in my own entries (so if you catch a \subset, please point it to me)
f
G > H G
p \ /_  ~ f(G)
\ / f ker f
G/ker f