# 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 A be a collection of sets.
S(R(A)) = ???
R(A) c S(A), hence S(R(A) c S(A)
A c R(A) c S(R(A)), hence S(A) c S(R(A))
S(R(A)) = S(A)
see page 35 in 1st measure
Let W c P(X). Let u : W -> |R* be additive.
(1) A, B, A\B :- W
(2) B c A
(3) u(A) :- |R
What can we conclude about the value of u(B)?
Proof:
1) B c A, hence A = B u (A\B) and B n (A\B) = O
2) u is additive, hence u(A) = u(B) + u(A\B)
3) Since u(A) :- |R, both u(B) and u(A\B) are real numbers.
Let W c P(X). Let u : W -> |R* be additive.
(1) B, A, A\B :- W
(2) B c A
(3) u(A) :- |R
What can we conclude about the value of u(A\B)?
Answer: u(A\B):-|R and u(A\B) = u(A) - u(B)
Proof:
1) B c A, hence A = B u (A\B) and B n (A\B) = O
2) u is additive, hence u(A) = u(B) + u(A\B)
3) Since u(A) :- |R, both u(B) and u(A\B) are real numbers.
4) u(A) - u(B) = u(A\B)
Let A[n] be a decreasing sequence of sets.
Prove that for every n:-|N
A[n] = A[1] \ ( U(k=1 to k=n-1) A[k]\A[k+1] )
2) page 53 in 2nd measure notebook
1) page 76 in 1st measure notebook
2) is better than 1)
x does not belong to A\B <=> [ ??? => ??? ]
x does not belong to A\B <=> [ x:-A => x:-B ]
inspiring line:{x does not belong to A\B <=> x:-AcB}
Let A[n] be a sequence of sets.
Prove that
//\\(n:-|N) A[n] = A[1] \ ( U(n:-|N) A[n] \ A[n+1] )
page 78 in 1st measure
Let W c P(X) be a s-ring.
Let y : W -> [0,oo] be countably additive and y(O)=0.
Let /\n:-|N A[n] :- W.
Suppose that the series y(A[n]) converges.
What can we conclude about the value of y(lim_sup A[n]) ?
y(lim_sup A[n]) = 0
page 54 in 2nd measure
Let a[n] be a sequence of complex numbers.
Suppose that the series a[n] converges.
What can we immediately conclude?
1) a[n] tends to zero
2) lim(k->oo) +(n=k to n=oo) a[n] = 0
Let W c P(X) be a s-ring.
Let y : W -> |R* be countably additive and y(O)=0.
Let A[n] be an increasing sequence of sets from W.
What can we conclude about the value y( U(n:-|N) A[n] ) ?
y( U(n:-|N) A[n] ) = lim(n->oo) y(A[n])
(this limit is proven to exist in the course of the proof)
page 157 in 1st measure
Let W c P(X) be a s-ring.
Let y : W -> |R* be countably additive and y(O)=0.
Let A[n] be an increasing sequence of sets from W.
What can we conclude about the limit of y(A[n]) ?
It exists and
lim(n->oo) y(A[n]) = y( U(n:-|N) A[n] )
hint: A[n] is increasing, hence A[n] = \\*//(k=1 to k=n) A[k]\A[k-1].
page 157 in 1st measure