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.

Suppose that 0 < x < y.
x/y ??? (x+1)/(y+1)
x/y < (x+1)/(y+1)
Let X be a set. Let d : XxX -> [0,oo].
(1) /\x,y,z:-X d(x,z) <= d(x,y) + d(z,y)
Let u:-X, 0 < r <= oo.
Let U = {x:-X : d(u,x) < r}.
What can we conclude about set U?
/\a:-U \/e>0 /\x:-X d(x,a)<e => x:-U
page 69 in gen top
Let K be a set and /\k:-K d[k] : XxX -> [0,oo].
Let W = { U c X : \/k:-K /\a:-U \/e>0 /\x:-X d[k](x,a)<e => x:-U }.
Let x:T->X be a net in X, and let u:-X.
(1) /\k:-K /\e>0 \/p:-T /\t>p d[k](x[t] , u) < e
What can we conclude?
net x W-converges to u
page 71 in gen top
Let K be a set and /\k:-K d[k] : XxX -> [0,oo].
(1) /\k:-K /\x:-X d[k](x,x)=0
(2) /\k:-K /\x,y:-X d[k](x,y) = d[k](y,x)
(3) /\k:-K /\x,y,z:-X d(x,z) <= d(x,y) + d(x,y)
Let W = { U c X : \/k:-K /\a:-U \/e>0 /\x:-X d[k](x,a)<e => x:-U }.
Let x:T->X be a net in X, which converges to u:-X.
page 71 in gen top
Let W be a s-ring in X.
Let y : W -> [0,oo] countably additive, y(O)=0.
In this context, what are the y-zero sets?
The symbol for the collection of all such sets,
let it be N(y).
N(y) = { EcX : \/A:-W EcA and y(A)=0 }
Recall that it is a s-ideal.
That is the subject matter of another item.
page 72 in 2nd measure
Imagine that a rectangle is written as a finite union of rectangles, which can overlap only along the edges. Decide if it must be true: The union of some two of these rectangles forms a rectangle.
FALSE
ADD
AOC
BBC
Let f be a real-valued function defined on a connected subset of |R.
Suppose that it is differentiable and that its derivative is bounded.
What can we conclude about this function?
It satisfies a Lipschitz condition.
Use Lagrange's mean value theorem to prove it.
page 156 in golden gate
Let f be a real-valued function defined on an open subset of |R.
Suppose that it is differentiable and Lipschitz.
What can we conclude about its derivative?
Its derivative is bounded.
Notice that f may be defined on a disconnected subset of |R.
page 156 in golden gate
Let f be a real-valued function defined on an open subset of |R.
Suppose that it is differentiable and that its derivative is bounded.
Does this function have to be Lipschitz?
If it's defined on a connected subset, then yes.
But generally, no.
See page 80 in OLDTIMER for an example.
Let R be a ring of sets in X.
Let W = { EcX : E:-R or X\E:-R }.
What can we conclude about W?
W is an algebra of sets in X
and W = K(R)
page 67 in 2nd measure