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 F be a s-algebra in X.
Let f : X -> |R* be measurable.
Prove:
/\a:-|R {x:-X : f(x)=a} :- F
page 134 in 1st measure
Let F be a s-algebra in X.
Let f[n] be a sequence of measurable functions X -> |R*.
Let g : X->|R* be defined by g(x) = sup(n:-|N) f[n](x).
What can we conclude about function g?
g is measurable
page 134 in 1st measure
Let F be a s-algebra in X.
Let f[n] be a sequence of measurable functions X -> |R*.
Let g : X->|R* be defined by g(x) = inf(n:-|N) f[n](x).
What can we conclude about function g?
g is measurable
page 134 in 1st measure
Let F be a s-algebra in X.
Let f[n] be a sequence of measurable functions X -> |R*.
Let g : X->|R* be defined by g(x) = lim_inf(n->oo) f[n](x).
What can we conclude about function g?
g is measurable
page 135 in 1st measure
Let F be a s-algebra in X.
Let f[n] be a sequence of measurable functions X -> |R*.
Let g : X->|R* be defined by g(x) = lim_sup(n->oo) f[n](x).
What can we conclude about function g?
g is measurable
page 135 in 1st measure
Decide if it's true:
/\a,b:-|R \/x,y:-|R [ x+y = a and x-y = b ]
true
x = (a+b)/2
y = (a-b)/2
f : |C -> |C
(1) f is diffable
(2) \/A:-|C /\z:-|C f'(z) = A*f(z)
(3) f(0) = 1
What can we conclude?
(4) /\a,b:-|C f(a+b) = f(a)*f(b)
hint: fix w and define g(z)=f(z)*f(w-z)
page 78 in OLDTIMER
Let A,B,C,D,E,F be non-empty sets.
Decide if it's true:
(A x C) u (B x D) = E x F ==> A=B or C=D
NO.
A={1}; B={1,2}; C={7}; D={7,8}; E={1,2}; F={7,8}
(A x C) u (B x D) = E x F and A!=B and C!=D
Let x > 0.
(x+2) / (x+1)^2 < ???
(x+2) / (x+1)^2 < 1/x
Let F be a s-algebra in X.
Let f[n] : X -> |R* be a sequence of measurable functions.
What can we say about this set:
{x:-X : \/A:-|R lim(n->oo) f[n](x) = A}
it belongs to F
page 83 in 2nd measure