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[n] be a sequence of sets.

Suppose that it converges to A.

Prove that every subsequence of {A[n]} also converges to A.

Suppose that it converges to A.

Prove that every subsequence of {A[n]} also converges to A.

A = lim_inf A[n] c lim_inf A[k(n)] c lim_sup A[k(n)] c lim_sup
A[n] = A

Prove that a countable intersection of sets from a s-ring
belongs to this s-ring.

//\\(n=1 to n=oo) [ A[n] ] = A[1] \ U(n=1 to n=oo) [ A[1] \
A[n] ]

Side remark:

Notice that this formula can be more general.

Countability is irrelevant.

The choice of the first set is also irrelevant.

Side remark:

Notice that this formula can be more general.

Countability is irrelevant.

The choice of the first set is also irrelevant.

Express the intersection A n B with the difference operation \.

A n B = A \ (A \ B)

A \ (A \ B) = ???

Express differently.

Express differently.

A \ (A \ B) = A n B

What is a ring of sets?

Let X be a set. Let R c P(X).

R is a ring of sets

iff

(1) R is nonempty

(2) A,B :- R => A u B :- R

(3) A,B :- R => A \ B :- R

R is a ring of sets

iff

(1) R is nonempty

(2) A,B :- R => A u B :- R

(3) A,B :- R => A \ B :- R

Let R be a ring of sets. Prove that A,B :- R => A n B :- R.

A n B = A \ (A \ B)

Let R be a ring of sets. Does it have to contain the empty set?

Yes.

In the definition of a ring we required that it is a non-empty collection of sets.

Hence some set belongs to it, say A. Then A\A belongs.

In the definition of a ring we required that it is a non-empty collection of sets.

Hence some set belongs to it, say A. Then A\A belongs.

Let R be a ring of sets. Prove that A,B :- R => A + B :- R.

____

What is a s-ring of sets?

Let X be a set. Let R c P(x).

R is a s-ring of sets

iff

(0) R is nonempty

(1) Countable union of sets from R belongs to R.

(2) A,B :- R => A \ B :- R

R is a s-ring of sets

iff

(0) R is nonempty

(1) Countable union of sets from R belongs to R.

(2) A,B :- R => A \ B :- R

Prove that if R is a s-ring of sets, then every countable
intersection of sets in R again belongs to R.

Use:

//\\(n=1 to n=oo) [ A[n] ] = A[1] \ U(n=1 to n=oo) [ A[1] \ A[n] ]

//\\(n=1 to n=oo) [ A[n] ] = A[1] \ U(n=1 to n=oo) [ A[1] \ A[n] ]