Math ASCII Notation Demo

Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

Let A,B be subsets of a metric space. Let p be a point in this space.
Suppose that for every ball with center p,
if this ball is contained in A, then it is not contained in B.
What can we conclude about point p?
p does not belong to Int( A n B )
Let G be an open subset of a metric space X.
Show that for every A c X,
G n Clo(A) is non-empty => G n A is non-empty.
______
Let G be an open subset of a metric space X.
Show that for every A c X,
G n A is empty => G n Clo(A) is empty.
_____
Let f[n]:E->C, g[n]:E->C be functional sequences.
Suppose that f converges uniformly to the zero function.
Suppose that g[n] is uniformly bounded.
What can we conclude about f[n]*g[n]?
f[n]*g[n] converges uniformly to the zero function.
Consider the series x^n / (1 - x^n).
Where does it converge? Consider x complex, |x| != 1.
|x|!=1 => [ |x|<1 <=> converges ]
X set, M c X.
W = {E c X : M\E is finite}
Describe R(W) - the ring generated by W.
R(W) = {E c X : M\E is finite v MnE is finite}
X set, W c P(X).
F : W -> [-oo,oo]
What does it mean that F is subadditive?
/\A,B:-W [ AuB :- W ==> F(AuB) <= F(A) + F(B) ]
X set, W c P(X).
F : W -> [-oo,oo]
What does it mean that F is finitely subadditive?
For every natural n, if A[1],...,A[n] belong to W, and if their union belongs to W,
then F(their union) <= +(k=1 to k=n) [ F(A[k]) ].
X set, W c P(X).
F : W -> [-oo,oo]
What does it mean that F is countably subadditive?
For every sequence A[n] of sets in W whose union is also in W,
we have F(their countable union) <= +(n=1 to n=oo) [ F(A[n]) ].
X set.
What is "the counting measure on X"?
F : P(X) -> [0,oo]
If A c X is finite, then F(A) = the number of elements of A.
If A c X is infinite, then F(A) = oo.
Notice that it is countably additive!