# 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.

Consider the p-dimensional Euclidean space |R^p.
Let W be the collection of all intervals in |R^p.
Define an additive function on W
which will eventually lead to the Lebesgue measure on |R^p.
For -oo < a <= b < oo,
if <a,b> is an interval in |R, then let m(<a,b>) = b - a.
If U = U[1] x ... x U[p], where U[k] is an interval in |R,
then let m(U) = m(U[1]) * ... * m(U[p]).
to see that this function is additive,
Consider the p-dimensional Euclidean space |R^p.
In the context of measure theory,
what are the elementary sets?
A subset of |R^p is elementary iff
it is a finite union of intervals in |R^p.
Let W be the collection of all intervals in |R^p.
Let E denote the collection of all elementary sets.
Then E = Ws.
page 102 in first measure
Consider the p-dimensional Euclidean space |R^p.
In the context of measure theory,
what are the properties of the collection of elementary sets
that follow immediately from its definition?
E is a ring of sets.
If W denotes the collection of all intervals in |R^p,
and E denotes the collection of all elementary sets,
then E = Ws = R(W) = the set of all finite disjoint unions of sets from W
Let A be a complex number.
What can we say about the limit of the sequence {cos(n*A)} ?
the sequence cos(n*A) does not tend to zero
page 189 in the palace notebook
Let A be an arbitrary set.
Let T be a proposition.
Let Y(a) be a proposition for every a:-A.
Find an equivalent formula for:
[ \/a:-A Y(a) ] => T
[ \/a:-A Y(a) ] => T <=> /\a:-A [ Y(a) => T ]
Let A be an arbitrary set.
Let T be a proposition.
Let Y(a) be a proposition for every a:-A.
Find an equivalent formula for:
/\a:-A [ Y(a) => T ]
/\a:-A [ Y(a) => T ] <=> [ \/a:-A Y(a) ] => T
Let X be a set. Let M be a subset of X.
Let W = { A c X : M c A }.
Describe R(W).
R(W) = { E c X : M c E or M n E = O }
page 56 in the 2nd measure notebook
Let X be a set. Let M be a subset of X.
Let W = { A c X : M c A }.
Describe S(W) - the s-ring generated by W.
S(W) = { E c X : M c E or M n E = O }
See page 142 in 2nd measure.
(page 57 in the 2nd measure notebook)
Let X be a set and let M be a countable subset of X.
Let W = { E c X : M \ E is finite }.
Describe S(W) - the s-ring generated by W.
S(W) = P(X)
page 58 in 2nd measure
(1) +(n=1 to n=oo) a[n] = oo
(2) +(n=1 to n=oo) b[n] = b, b is a real number
What can we conclude?
+(n=1 to n=oo) a[n]+b[n] = oo
page 193 in 1st measure