On Thu, 27 Mar 2008, sto wrote:
> This is a question about symbols and notation, so it is hard to post on
> a text forum, but it has been driving me crazy so here goes. In the
> book "Measure, Integral, and Probability" by Capinski and Kopp the
> authors define the set inclusion notation "A subset B" (expressed here
> in LaTeX) to mean x in A => x in B. The authors clarify that they do
It does not. A subset B means
for all x, (x in A ==> x in B)
or
for all x in A, x in B
> not restrict themselves to a proper subset by this notation, ie A subset
> B does not exclude the possibility that A = B. This is the usual subset
> symbol without the line underneath.
>
Don't use LaTex. Common usage I've come to is
a^n a to the n-th
a_n a sub n
A subset B A c B, A C B, A < B are to be discouraged
A proper subset B
A /\ B, A \/ B A intersection B, A union B
int A, cl A, bd A interior, closure, boundary
integral(a,b) f(x) dx preferable to calculator style
sum(j=0,n) a^j sum on j from 0 to n
lim(x->a) f(x)
> Then throughout the book they interdisperse this notation with the
> notation A subseteq B (the subset symbol with a line underneath) in
> various definitions and theorems. The bizarre thing is that given their
> convention for A subset B, it seems to me that the two notations should
> mean *exactly* the same thing. This creates confusion (at least for me)
> because if A subseteq B denotes the same thing as A subset B, then why
> are they using both notations instead of one? If, on the other hand,
> the two notations mean different things, and if A subset B includes the
> possibility that A=B, then what can A subseteq B possibly mean?
>
A proper subset B when A subset B and A /= B.
As proper subset is much less used than subset,
the expersion subset prevails to include equality.
> I checked the errata sheet in the springer verlag website and there is
> nothing there about this. Has anyone worked through the proofs in this
> book and figured this out?
>
The distinction is essential. For example
A set S is infinite iff there's a proper subset
P which is equinumerous to S, ie |P| = |S|.
Oh the other hand
A set S is finite or infinite iff there's a subset
P which is equinumerous to S, ie |P| = |S|.
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