# 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 T be the set of all finite subsets of |N.
For a,b:-T, a>b <=> b is contained in a
(T,>) is a directed set.
Let t[n] = (-1)^(n+1) * (1/n), for n:-|N.
For F:-T, let S(F) = the finite sum of t[n] where n:-F.
Does the net S(F) converge?
NO.
page 169 in "Topological Spaces, From Distance to Neighborhood"
What is a singleton?
A singleton is a set that has exactly one element.
{x}
What do you call a set that has exactly one element?
a singleton ['siN g.l t.n]
Consider the complex function Re(z).
Prove that it is nowhere diffable.
=
a,b:-[-pi,pi] ; cos(a)=cos(b)
Show that a=b or a=(-b).
hint: cos is 1-1 on [0,pi] and cos(|a|) = cos(|b|)
Let t[n] be a sequence satisfying -pi<=t[n]<=pi. Let -pi<t<pi. Suppose that cos(t[n])->cos(t) and sin(t[n])->sin(t). What can we conclude?
t[n]->t
page 38 in OLDTIMER
Express differently:
max(x,y) = ???
max(x,y) = ( x + y + |x - y| ) / 2
Express differently
min(x,y) = ???
min(x,y) = ( x + y - |x - y| ) / 2
Let A[n] be a sequence of sets contained in X.
Prove that the two conditions are equivalent.
(1) A[n] converges
(2) /\x:-X [ {n:-|N : x:-A[n]} is finite or {n:-|N : not x:- A[n]} is finite} ]
page 20 first measure theory
Let A[n] be a sequence of sets contained in X. Let A c X.
Prove that the three conditions are equivalent.
(1) A[n] -> A
(2) A[n] + A -> O
(3) lim_sup (A[n] + A) = O
page 21 measure theory