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

Describe K(W) - the algebra generated by W.

Describe K(W) - the algebra generated by W.

K(W) = { A c X : A is finite or 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 d(W) - the s-algebra generated by W.

Describe d(W) - the s-algebra generated by W.

d(W) = { A c X : A is countable or X\A is countable }

page 45 in 1st measure

page 45 in 1st measure

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

Describe M(W) - the monotone class generated by W.

Describe M(W) - the monotone class generated by W.

M(W) = W

Notice that W is itself a monotone class.

page 45 in 1st measure

Notice that W is itself a monotone class.

page 45 in 1st measure

Let D be an open subset of |R^2. Let f:D->|R. Let a,b:-|R, a<b.

(1) the partial derivatives fx, fy exist on D and they are continuous on D

(2) u : ]a,b[ -> U, U c |R

(3) v : ]a,b[ -> V, V c |R

(4) u and v are continuously diffable on ]a,b[.

(5) UxV c D.

(1) the partial derivatives fx, fy exist on D and they are continuous on D

(2) u : ]a,b[ -> U, U c |R

(3) v : ]a,b[ -> V, V c |R

(4) u and v are continuously diffable on ]a,b[.

(5) UxV c D.

F is diffable on ]a,b[.

F'(t) = fx(u(t),v(t))*u'(t) + fy(u(t),v(t))*v'(t)

page 50 in OLDTIMER

F'(t) = fx(u(t),v(t))*u'(t) + fy(u(t),v(t))*v'(t)

page 50 in OLDTIMER

Let X be a non-empty set. Let W c P(X).

Suppose that M(W) is a ring of sets.

What is the relation between M(W) and S(W)?

Suppose that M(W) is a ring of sets.

What is the relation between M(W) and S(W)?

It is always true that M(W) c S(W).

To prove that S(W) c M(W), we show that M(W) is a s-ring. Use:

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

(2) The union of an increasing sequence of sets from K belongs to K.

imply that

(3) The union of any sequence of sets from K belongs to K.

To prove that S(W) c M(W), we show that M(W) is a s-ring. Use:

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

(2) The union of an increasing sequence of sets from K belongs to K.

imply that

(3) The union of any sequence of sets from K belongs to K.

Express differently:

A u U(t:-T) B[t] = ???

A u U(t:-T) B[t] = ???

A u U(t:-T) B[t] = U(t:-T) AuB[t]

Express differently:

U(t:-T) AuB[t] = ???

U(t:-T) AuB[t] = ???

U(t:-T) AuB[t] = A u U(t:-T) B[t]

Express differently:

A n U(t:-T) B[t] = ???

A n U(t:-T) B[t] = ???

A n U(t:-T) B[t] = U(t:-T) AnB[t]

Express differently:

U(t:-T) AnB[t] = ???

U(t:-T) AnB[t] = ???

U(t:-T) AnB[t] = A n U(t:-T) B[t]

Express differently:

A n //\\(t:-T) B[t] = ???

A n //\\(t:-T) B[t] = ???

A n //\\(t:-T) B[t] = //\\(t:-T) AnB[t]