# 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 (X,d) be a metric space. Let (Y,g) be a complete metric space. Let A be a dense subset of X. Consider a Lipschitz function f:A->Y. Can this function be extended to f:X->Y and still be Lipschitz?
YES
Prove that for every metric space (X,d) there exist:
(1) a complete metric space (X',d')
(2) an isometry f:X->X'
such that f(X) is dense in X'.
page 53 in golden gate
or
page 138 in the book "Topological Spaces - From Distance to Neighborhood"
Let {A[n]} be a sequence of sets. Define the lower limit of this sequence.
\\//(k=1 to k=oo) //\\ (n=k to n=oo) A[n]
x belongs to the lower limit of a sequence of sets
iff
x belongs to almost all sets from this sequence
Let {A[n]} be a sequence of sets. Define the upper limit of this sequence.
//\\(k=1 to k=oo) \\// (n=k to n=oo) A[n]
x belongs to the upper limit of a sequence of sets
iff
x belongs to infinitely many sets from this sequence
What does it mean that a sequence of sets converges?
The upper limit of this sequence is contained in the lower limit.
For all x, if x belongs to infinitely many sets of this sequence, then it belongs to almost all sets of this sequence.
What does it mean that a sequence of sets is increasing?
{A[n]} sequence of sets
/\n:-N [ A[n] c A[n+1] ]
(A constant sequence is increasing.)
What does it mean that a sequence of sets is decreasing?
{A[n]} sequence of sets
/\n:-|N [ A[n+1] c A[n] ]
(A constant sequence is decreasing.)
What does it mean that a sequence of sets is monotone?
['mo n.. tOun]
It's increasing or it's decreasing.
See the previous two items.
A + B = (A\B) u (B\A)
What do you call this operation on sets?
the symmetric difference of two sets A and B
[si 'met rik]
What is a disjoint class of sets?
Every two sets of this class are disjoint.
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2001.02.06
This definition is stupid.