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.

Prove that the intersection of any number of rings of sets is again a ring of sets.
______
Prove that the intersection of any number of s-rings of sets is again a s-ring of sets.
_________
Let X be a nonempty set. Let W c P(X). What is "the ring generated by W"?
The smallest ring containing W.
Such a ring exists.
It is the intersection of all rings containing W.
We denote it R(W).
Let X be a nonempty set. Let W c P(X). What is "the s-ring generated by W"?
The smallest s-ring containing W.
Such a s-ring exists.
It is the intersection of all s-rings containing W.
We denote it S(W).
What is an algebra of sets? Give a minimal definition.
Let X be a set. Let R c P(X).
R is an algebra of sets
iff
(1) X :- R
(2) A,B :- R => A u B :- R
(3) B :- R => X \ B :- R
What is a s-algebra of sets?
Let X be a set. Let W c P(X).
W is a s-algebra of sets
iff
(1) X :- W
(2) /\n:-|N A[n]:-W |=> U(n:-|N)_A[n] :- W
(3) B:-W => X\B:-W
Consider a collection of algebras of subsets of X.
X belongs to all the algebras.
Prove that the intersection of this collection is an algebra of subsets of X.
page 33 in 1st measure
Consider a collection of s-algebras of subsets of X.
X belongs to all the s-algebras.
Prove that the intersection of this collection is a s-algebra of subsets of X.
page 34 in first measure
Let X be a set. Let W c P(X).
What is "the algebra generated by W"?
The smallest algebra containing W.
Such an algebra exists.
It is the intersection of all algebras containing W.
We denote it K(W).
Let X be a set. Let W c P(X).
What is "the s-algebra generated by W"?
The smallest s-algebra containing W.
Such a s-algebra exists.
It is the intersection of all s-algebras containing W.
We denote it d(W).