Math ASCII Notation Demo

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,d) be a metric space. Let A c X.
What does it mean that A is of the first category?
A set of the first category is a countable union of rare sets.
(The closure of a rare set contains no ball.)
(Every rare set is of the first category.)
a set of the first category = a meager set
Let (X,d) be a metric space. A c B c X.
If B is meager in X, then A is meager in X.
Decide if it's true. Prove your answer.
TRUE
Let (X,d) be a metric space. A c B c X.
If A is meager in X, then B is meager in X.
Decide if it's true. Prove your answer.
FALSE.
Let (X,d) be a metric space. A c B c X.
If A is rare in B, then A is rare in X.
Decide if it's true. Prove your answer.
TRUE
To prove this, use (p->q) <== (-q -> -p).
page 178 in the palace notebook
Let (X,d) be a metric space. A c B c X.
If A is rare in X, then A is rare in B.
Decide if it's true. Prove your answer.
FALSE.
Let X = |R, and let A be the Cantor.
Then A is rare in X.
Let B=A, then AcB but A is not rare in B.
Let (X,d) be a metric space. A c B c X.
If A is meager in B, then A is meager in X.
Decide if it's true. Prove your answer.
TRUE.
Use: If A is rare in B, then A is rare in X.
Let (X,d) be a metric space. A c B c X.
If A is meager in X, then A is meager in B.
Decide if it's true. Prove your answer.
FALSE
Let A=B={1}. Let X=|R.
Then A is meager in X, but A is not meager in B.
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.
Suppose that both the sequences are uniformly bounded.
Is it true that f[n]*g[n] converges uniformly to f*g?
YES
page 178 in the palace notebook
Let (X,d) be a bounded metric space.
Let H be the set of all closed subsets of X except the empty set.
Define h:HxH->R by h(A,B) = sup{|d(x,A) - d(x,B)| : x:-X}.
Prove that (H,h) is a metric space.
This is called the Hausdorff metric space.
page 179 in the palace notebook
Let (X,d) be a bounded metric space.
Let H be the set of all closed subsets of X except the empty set.
What is the definition of the Hausdorff metric space?
(Just recall the definition without proving that it is a metric space.)
h:HxH->R by h(A,B) = sup{|d(x,A) - d(x,B)| : x:-X}.
(H,h) is a metric space, see the previous item.