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 B[1],B[2],B[3],... be an increasing sequence of sets.

Give an interesting way of writing the union of this sequence.

Give an interesting way of writing the union of this sequence.

\\//(n=1 to n=oo) B[n] = \\*//(n=1 to n=oo) B[n]\B[n-1],

where B[0] = O.

Notice that the equality always holds, even if the sequence is not increasing.

Increasingness is necessary to show the disjointness of the right-hand union.

where B[0] = O.

Notice that the equality always holds, even if the sequence is not increasing.

Increasingness is necessary to show the disjointness of the right-hand union.

Let W be a collection of sets.

(1) The union of two sets from W belongs to W.

(2) The union of an inreasing sequence of sets from W belongs to W.

What more can we say about W?

(1) The union of two sets from W belongs to W.

(2) The union of an inreasing sequence of sets from W belongs to W.

What more can we say about W?

The union of any sequence of sets from W belongs to W.

Let A[n] be a sequence of sets.

Express more simply:

U(n:-N) [ A[n] \ (U(k=1 to n-1) A[k] ) ] = ???

Express more simply:

U(n:-N) [ A[n] \ (U(k=1 to n-1) A[k] ) ] = ???

= U{A[n]}

page 41 in 1st measure

page 41 in 1st measure

Suppose that a collection of sets is monotone and is a ring.
What can we further say about this collection?

It is a s-ring.

(1) The union of two sets from W belongs to W.

(2) The union of an inreasing sequence of sets from W belongs to W.

imply that

(1) The union of two sets from W belongs to W.

(2) The union of an inreasing sequence of sets from W belongs to W.

imply that

Suppose that a collection of sets is monotone and is an
algebra. What can we further say about this collection?

It is a s-algebra.

(1) The union of two sets from W belongs to W.

(2) The union of an inreasing sequence of sets from W belongs to W.

imply that

(1) The union of two sets from W belongs to W.

(2) The union of an inreasing sequence of sets from W belongs to W.

imply that

Let X be a set. Consider a collection of monotone classes of
subsets of X. Does the intersection of this collection have to
be a monotone class?

YES.

page 44 in 1st measure

page 44 in 1st measure

Let X be a nonempty set. Let W c P(X).

What is "the monotone class generated by W"?

What is "the monotone class generated by W"?

The smallest monotone class containing W.

Such a class exists.

It is the intersection of all monotone classes containing W.

page 44 in 1st measure

Such a class exists.

It is the intersection of all monotone classes containing W.

page 44 in 1st measure

Let X be a nonempty set. Let W c P(X).

What are the inclusion relations between R(W),S(W), K(W), d(W), M(W) ?

What are the inclusion relations between R(W),S(W), K(W), d(W), M(W) ?

1) R(W) c S(W)

2) K(W) c d(W)

3) R(W) c K(W)

4) S(W) c d(W)

5) M(W) c S(W)

6) M(W) c d(W)

2) K(W) c d(W)

3) R(W) c K(W)

4) S(W) c d(W)

5) M(W) c S(W)

6) M(W) c d(W)

Let X be a non-empty set. Let W={ {x} : x:-X }.

Describe R(W) - the ring generated by W.

Describe R(W) - the ring generated by W.

R(W) = { A c X : A is finite }

page 45 in 1st measure

page 45 in 1st measure

Let X be a non-empty set. Let W={ {x} : x:-X }.

Describe S(W) - the s-ring generated by W.

Describe S(W) - the s-ring generated by W.

S(W) = { A c X : A is countable }

page 45 in 1st measure

countable = equinumerous with a subset of |N

page 45 in 1st measure

countable = equinumerous with a subset of |N