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.

??? c (A u B) x (M u N)
(A x M) u (B x N) c (A u B) x (M u N)
Is this generally true?
(A u B) x (M u N) c (A x M) u (B x N)
No.
A = N = {1}
M = B = O
X, W c P(X), W is a semi-ring in X
Y, V c P(Y), V is a semi-ring in Y
Prove that { A x B : A:-W , B:-V } is a semi-ring in XxY.
page 67 in 1st measure
Let A,B be two semi-rings in X.
Does their intersection (A n B) have to be a semi-ring in X?
NO.
A = { O, {1,2,3}, {1}, {2,3} }
B = { O, {1,2,3}, {1}, {2}, {3} }
page 69 in 1st measure
In the extended real number system, how is multiplication defined?
(1) /\x,y:-|R* x*y = y*x
(2) 0*oo = 0
(3) /\x:-|R* [ x>0 ==> x*oo = oo ]
(4) /\a,b,c:-|R* (a*b)*c = a*(b*c)
(5) (-oo) = (-1)*oo
page 70 in 1st measure
In the extended real number system,
why can't we define: oo+(-oo)=0?
Because then we could prove that oo+(-oo)=7 by using associativity.
1) oo+(-oo)=0
2) 7+[ oo+(-oo) ] = 7 + 0
3) [7 + oo] + (-oo) = 7
4) oo + (-oo) = 7
Let X be a set, and W c P(X).
What does it mean that function T:W->[-oo,oo] is additive?
/\B,A:-W [ BuA:-W and BnA=O ==> T(B u A) = T(B) + T(A) ]
Notice that if T is additive on W and O:-W,
then T(O)=0 or T(O)=oo or T(O)=-oo.
Let X be a set, W c P(X), W is a ring.
Let T:W->|R* be additive.
Is it possible that T(A)=oo and T(B)=-oo for some A,B:-W ?
NO.
page 71 in 1st measure
f : |C -> |C
(1) f(0) = 1
(2) /\w,z:-|C f(w+z) = f(w)*f(z)
(3) \/a:-|C lim(z->0) (f(z)-1) / z = a
What can we conclude about function f?
f is diffable and
/\z:-|C f'(z) = a*f(z)
page 75 in OLDTIMER
A,B c X
1) A n B = O <=> ???
2) A n B = O <=> ???
1) A n B = O <=> A c X\B
2) A n B = O <=> B c X\A