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,B be subsets of X. If GcX, then 1(G) denotes the
characteristic function of G.

Express differently:

( 1(A)<=1(B) ) <=> ???

Express differently:

( 1(A)<=1(B) ) <=> ???

( 1(A)<=1(B) ) <=> A c B

Let x>0, y>0.

Complete without proof:

Arg(x+iy) = arctan(....)

Complete without proof:

Arg(x+iy) = arctan(....)

Arg(x+iy) = arctan(y/x)

page 91 in OLDTIMER

page 91 in OLDTIMER

Let x>0, y>0.

Complete without proof:

Arg(x+iy) = arccos( ??? )

Complete without proof:

Arg(x+iy) = arccos( ??? )

Arg(x+iy) = arccos(x / sqrt(x*x+y*y))

You can find the proof on pages 91,95 in OLDTIMER.

You can find the proof on pages 91,95 in OLDTIMER.

Show a function that satisfies:

(1) f:D->|R; D is an open subset of |R^2

(2) partial derivatives fxy and fyx exist at (x0,y0)

(3) fxy(x0,y0) != fyx(x0,y0)

(1) f:D->|R; D is an open subset of |R^2

(2) partial derivatives fxy and fyx exist at (x0,y0)

(3) fxy(x0,y0) != fyx(x0,y0)

f(0,0) = 0

f(x,y) = xy(x^2-y^2) / (x^2+y^2)

fxy(0,0) = -1

fyx(0,0) = 1

f(x,y) = xy(x^2-y^2) / (x^2+y^2)

fxy(0,0) = -1

fyx(0,0) = 1

Let R c P(X) be a ring of sets. Let A c X.

M(R) denotes the monotone class generated by R.

Show that { B:-M(R) : AuB,A\B,B\A :- M(R) } is a monotone class.

M(R) denotes the monotone class generated by R.

Show that { B:-M(R) : AuB,A\B,B\A :- M(R) } is a monotone class.

page 51 in 1st measure

Let R c P(X) be a ring of sets. Let A:-R.

M(R) denotes the monotone class generated by R.

Prove that R is contained in { B:-M(R) : AuB,A\B,B\A :- M(R) }.

M(R) denotes the monotone class generated by R.

Prove that R is contained in { B:-M(R) : AuB,A\B,B\A :- M(R) }.

page 52 in 1st measure

Let R c P(X) be a ring of sets. Let A:-R.

M(R) denotes the monotone class generated by R.

Prove that M(R) is contained in { B:-M(R) : AuB,A\B,B\A :- M(R) }.

M(R) denotes the monotone class generated by R.

Prove that M(R) is contained in { B:-M(R) : AuB,A\B,B\A :- M(R) }.

Step1: R is contained in { B:-M(R) : AuB,A\B,B\A :- M(R) }.

Step2: { B:-M(R) : AuB,A\B,B\A :- M(R) } is a monotone class.

See the two previous items.

page 53 in 1st measure

Step2: { B:-M(R) : AuB,A\B,B\A :- M(R) } is a monotone class.

See the two previous items.

page 53 in 1st measure

Let R c P(X) be a ring of sets.

M(R) denotes the monotone class generated by R.

Let A belong to M(R).

Prove that M(R) is contained in { B:-M(R) : AuB,A\B,B\A :- M(R) }.

M(R) denotes the monotone class generated by R.

Let A belong to M(R).

Prove that M(R) is contained in { B:-M(R) : AuB,A\B,B\A :- M(R) }.

First show that R is contained in { B:-M(R) : AuB,A\B,B\A :-
M(R) }.

You have to use the previous item to do this.

page 53 in 1st measure

You have to use the previous item to do this.

page 53 in 1st measure

Solve the equation: exp(z) = 2 + 2*i.

Re(z) = log(2*sqrt(2))

Im(z) = PI/4 + k*2*PI

Im(z) = PI/4 + k*2*PI

Let R c P(X) be a ring of sets.

M(R) denotes the monotone class generated by R.

S(R) denotes the s-ring generated by R.

Prove that M(R) = S(R).

M(R) denotes the monotone class generated by R.

S(R) denotes the s-ring generated by R.

Prove that M(R) = S(R).

0) Define for AcX, M[A] = { B:-M(R) : AuB,A\B,B\A :- M(R) }.

1) /\AcX M[A] is a monotone class.

2) /\A:-R R c M[A]

3) /\A:-R M(R) c M[A]

4) /\A:-M(R) R c M[A]

5) /\A:-M(R) M(R) c M[A]

6) Conclude that M(R) is a ring.

1) /\AcX M[A] is a monotone class.

2) /\A:-R R c M[A]

3) /\A:-R M(R) c M[A]

4) /\A:-M(R) R c M[A]

5) /\A:-M(R) M(R) c M[A]

6) Conclude that M(R) is a ring.