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 W c P(X), A c X.
Let A^ be the set of all points which are W-adherent to A.
Let Clo(A) = //\\ {FcX: F is W-closed and A c F}.
What can we say about these two sets?
A^ = Clo(A)
page 65 in gen top
Let W c P(X), A c X, x :- X.
What does it mean that x is sequentially W-adherent to A ?
There exists a sequence in A, which W-converges to x.
Let W c P(X), A c X.
What does it mean that A is sequentially W-closed ?
(1) Every point, that is sequentially W-adherent to A, belongs to A.
(2) If a sequence contained in A W-converges to x, then x belongs to A.
These two are equivalent.
page 66 in gen top
Let X be a set, and W c P(X), and A c X.
Let As denote the set of all points which are sequentially W- adherent to A.
Does As have to be sequentially W-closed?
NO.
page 67 in gen top
Let W be a ring. Let y : W -> |R* be additive. Let A,B :- W.
Suppose that B c A and y(A):-|R.
Is it possible that y(B) = oo or that y(B) = -oo ?
NO.
1) Since B c A, we have A = B u (A\B) disjointly.
2) Since W is a ring and y is additive on W, we have y(A) = y(B) + y(A\B).
3) Since y(A):-|R, both y(B) and y(A\B) must be real.
4) Hence y(B) :- |R.
page 75 in 1st measure
Let E be the collection of all elementary subsets of |R^p.
Let y : E -> [0,oo) be additive and regular.
Let y* be the corresponding outer measure on |R^p.
Define the collection MF(y) of all finitely y-measurable subsets of |R^p.
MF(y) = { A c |R^p : \/{A[n]}cE lim(n->oo) y(A[n]+A) = 0 }
page 117 in 1st measure
Let E be the collection of all elementary subsets of |R^p.
Let y : E -> [0,oo) be additive and regular.
Let y* be the corresponding outer measure on |R^p.
Let MF(y) denote the collection of all finitely y-measurable subsets of |R^p.
Prove that it is a ring.
You can use the theorem from the next item.
page 118 in 1st measure
page 64 in 2nd measure
Let X be a set, and let Y : P(X) -> [0,oo] satisfy:
(1) Y(O) = 0.
(2) A c B c X ==> Y(A) <= Y(B).
(3) Y(A u B) <= Y(A) + Y(B).
Let M be a ring in X.
Let F = { A c X : \/{A[n]}cM lim(n->oo) Y(A[n] + A) = 0 }.
Can we conclude that F is a ring?
F is a ring
page 64 in 2nd measure
page 118 in 1st measure
Let E be the collection of all elementary subsets of |R^p.
Let y : E -> [0,oo) be additive and regular.
Let y* be the corresponding outer measure on |R^p.
Let MF(y) denote the collection of all finitely y-measurable subsets of |R^p.
/\B,A:-MF(y) y*(A u B) + y*(A n B) = ???
/\B,A:-MF(y) y*(A u B) + y*(A n B) = y*(A) + y*(B)
page 118 in 1st measure
Let E be the collection of all elementary subsets of |R^p.
Let y : E -> [0,oo) be additive and regular.
Let y* be the corresponding outer measure on |R^p.
Let MF(y) denote the collection of all finitely y-measurable subsets of |R^p.
Define the collection M(y) of all y-measurable subsets of |R^p.
M(y) = the collection of countable unions of sets from MF(y)
page 117 in 1st measure