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.

A,B c X
Express differently
X \ (A \ B) = ???
X \ (A \ B) = (X \ A) u B
A,B c X
Express differently
(X \ A) u B = ???
(X \ A) u B = X \ (A \ B)
Let X be a nonempty set. Let W be a collection of finite subsets of X. What kind of collection is W?
W is an ideal.
W is a ring of sets.
Let X be a nonempty set. Let M c X. Let W be the collection of all A c X such that MnA is finite or M\A is finite. What kind of collection is W?
It is an algebra of sets.
PROOF:
A,B:-W
If MnA and MnB are both finite, then Mn(AuB) is finite, hence AuB:-W.
If M\A is finite, then M\(AuB) c M\A, hence M\(AuB) is finite, hence AuB:-W.
Let X be a nonempty set. Let W be the collection of all countable subsets of X. What kind of collection is W?
It is a s-ring of sets.
It is a s-ideal.
Let X be a nonempty set. Let M c X. Let W be the collection of all A c X such that MnA is countable or M\A is countable. What kind of collection is W?
It is a s-algebra of sets.
How do I choose to understand the word "countable" ?
countable = empty or finite or equinumerous with the set of natural numbers
Is it true that every countable union of sets in a ring may be written as a disjoint countable union of sets in the ring?
U(n:-|N) A[n] = \\*//(n:-|N) [ A[n] \ (U(k=1 to k=n-1) A[k]) ]
where A[n] = O
page 41 in 1st measure
(1) h : |R -> |R is diffable
(2) h(0) = 0
(3) /\x:-|R h'(x) + h(x)*cos(x) = cos(x)
Find the set of all such functions.
There is one and only one such function.
h(x) = 1 - exp(-sin(x))
hint: multiply both sides of (3) by exp(sin(x))
Let A[n] be a sequence of sets.
Express its union as the union of an increasing sequence of sets.
U{A[n]} = U(n:-N) [ U(k=1 to k=n) A[k] ]