# set

## Primary tabs

Defines:
contains, subset, proper subset
Type of Math Object:
Definition
Major Section:
Reference

## Mathematics Subject Classification

### \subset vs. \subseteq

Hi

I think this entry defines \subset and \subseteq in a non-standard 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
non-standard interpretation.

I would suggest to only define \subset, and in a remark-section 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 less-than and less-than-or-equal 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 on-line in PDF format: http://www.thoralf.uwaterloo.ca/htdocs/ualg.html

A couple more I have to hand are "Algebra in the Stone-Cech 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