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 be a complete metric space, with no isolated points. Let A be a dense subset of X. Suppose that A is a countable intersection of open sets. What can we conclude about A?
A is uncountable
page 48 in OLDTIMER
Let X be a metric space with no isolated points. Let A be a countable dense subset of X. Suppose that A is a countable intersection of open sets. What can we conclude about the metric space X?
X is incomplete
page 48 in OLDTIMER
Let X be a complete metric space, with no isolated points. Let A be a countable subset of X, which is a countable intersection of open sets. What can we conclude about A?
A is not dense
page 48 in OLDTIMER
Let X be a complete metric space. Let A be an infinitely countable dense subset of X, which is a countable intersection of open sets. Is this setup possible?
It is possible. For example:
Let X be {0} u {1/n : n:-N}.
Let A = X.
Let A be a countable intersection of itself.
Or, let A = X\{0}.
Let X be a complete metric space with no isolated points. Let A be a countable dense subset of X, which is a countable intersection of open sets. Is this setup possible?
NO.
page 48 in OLDTIMER
inf {x + 1/x : x>0} = ???
inf {x + 1/x : x>0} = 2
X set, W c P(X).
(1) X :- W
(2) A,B :- W => A u B :- W
(3) A :- W => X\A :- W
Prove that
(4) A,B :- W => A n B :- W
X \ (A n B) = (X\A) u (X\B) belongs, hence (4) is OK.
A \ B = A n (X\B) belongs, hence (5) is OK.
Let f be a complex function defined on an open set. Suppose that this function is diffable at a point. Prove that at this point function Re(f) has a partial derivative with respect to the first variable.
page 18 in OLDTIMER
Let f be a complex function defined on an open set. Suppose that this function is diffable at a point. Prove that at this point function Im(f) has a partial derivative with respect to the second variable.
page 18 in OLDTIMER
Let m be a natural number. Let z be a complex number with |z| < 1.
Express differently:
+(n=0 to n=oo) [ z^n * (n+m)! / n!*m! ] = ???
= 1 / (1-z)^(m+1)
page 190 in the palace notebook