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.

Express differently:

//\\(t:-t) AnB[t] = ???

//\\(t:-t) AnB[t] = ???

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

Express differently:

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

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

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

Express differently:

//\\(t:-T) AuB[t] = ???

//\\(t:-T) AuB[t] = ???

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

Express differently:

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

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

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

Express differently:

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

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

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

We want to define a metric on the interval [0,1).

We want the metric to be similar to the Euclidean metric.

Is it possible to define such a metric so that [0,1) is a complete metric space?

We want the metric to be similar to the Euclidean metric.

Is it possible to define such a metric so that [0,1) is a complete metric space?

Yes.

d(x,y)=|x/(1-x) - y/(1-y)|

f:[0,1)->[0,oo), f(x)=x/(1-x), f is a continuous bijection

d(x,y)=|x/(1-x) - y/(1-y)|

f:[0,1)->[0,oo), f(x)=x/(1-x), f is a continuous bijection

X set; F:P(X)->[0,oo]; F(O)=0; E,FcX; (G'=X\G)

(a) /\AcX F(A) = F(A n E) + F(A n E')

(b) /\AcX F(A) = F(A n F) + F(A n F')

Prove something interesting for the set E u F.

(a) /\AcX F(A) = F(A n E) + F(A n E')

(b) /\AcX F(A) = F(A n F) + F(A n F')

Prove something interesting for the set E u F.

/\AcX F(A) = F( A n (EuF) ) + F( A n (EuF)' )

page 39 in 2nd measure

page 39 in 2nd measure

Let f[n], g[n] be sequences of functions X->|C. Let f,g : X->|C.

Suppose that f[n] and g[n] converge uniformly to f and g respectively.

Can we conclude that f[n]*g[n] converges uniformly to f*g?

Suppose that f[n] and g[n] converge uniformly to f and g respectively.

Can we conclude that f[n]*g[n] converges uniformly to f*g?

NO.

Let X be |R. Let f[n](x) = 1/n. Let f(x)=0. Let g[n](x) = g(x) = x.

Let X be |R. Let f[n](x) = 1/n. Let f(x)=0. Let g[n](x) = g(x) = x.

Let (X,d) be a metric space. Let A c X.

Consider the set Fr(A) = Clo(A) \ Int(A) = Clo(A) n Clo(X\A).

Is it possible that Fr(A) contains a ball?

Consider the set Fr(A) = Clo(A) \ Int(A) = Clo(A) n Clo(X\A).

Is it possible that Fr(A) contains a ball?

YES.

It can even be an open set.

It can even be the whole space.

Consider |Q as a subset of |R with the Euclidean metric.

Then Fr(|Q) = |R.

It can even be an open set.

It can even be the whole space.

Consider |Q as a subset of |R with the Euclidean metric.

Then Fr(|Q) = |R.

sup { K > 0 : /\a,b:-R K*(|a|+|b|) <= sqrt(a*a+b*b) } = ???

1/sqrt(2)