|
| Play Piano Online |
| [Apronus Home] [Mathematics] |
|
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 f : X -> Y. Let A,B c Y. Suppose that A c f(X) and f-1(A) c f-1(B). Does this imply that A c B ? |
| YES page 87 in 2nd measure |
| Let f : X -> Y. Prove that /\BcY f-1(B) = f-1(B n ???) |
| /\BcY f-1(B) = f-1(B n f(X)) page 97 in 2nd measure |
| Let f : X -> Y. Let W c P(Y). Let K = { f-1(B) : B:-W }. Suppose that W is monotone. Does K have to be monotone, too? |
| NO page 88 in 2nd measure f : {1,2,3,4,...} -> {0,1,-1,2,-2,3,-3,4,-4,...}; f(n)=n W = { {1,-1}, {1,2,-2}, {1,2,3,-3}, {1,2,3,4,-4}, ... } K = { f-1(B) : B:-W } = { {1}, {1,2}, {1,2,3}, {1,2,3,4}, ... } W is monotone but K is not monotone |
| Let F a s-algebra in X. Let f : X -> |R* be measurable. Prove that /\Gc|R [ G is open => ??? ] |
| /\Gc|R [ G is open => f-1(G) :- F ] page 89 in 2nd measure |
| Let F be a s-algebra in X. Let f : X -> |R* be measurable. Prove that /\Gc|R [ ??? => f-1(G) :- F ] |
| (1) /\Gc|R [ G is open => f-1(G) :- F ] (2) /\Gc|R [ G is Borel => f-1(G) :- F ] page 89,90 in 2nd measure In order to prove (2) we use (1). To prove (1), we recall that every open set |
| Let f : X -> Y. Let M c P(X). Let W = { A c Y : f-1(A) :- M }. Suppose that M is a ring. What can we conclude about W ? |
| W is also a ring page 89 in 2nd measure |
| Let f : X -> Y. Let M c P(X). Let W = { A c Y : f-1(A) :- M }. Suppose that M is a s-ring. What can we conclude about W ? |
| W is also a s-ring page 89 in 2nd measure |
| Let f : X -> Y. Let M c P(X). Let W = { A c Y : f-1(A) :- M }. Suppose that M is an algebra in X. What can we conclude about W ? |
| W is an algebra in Y page 89 in 2nd measure |
| Let f : X -> Y. Let M c P(X). Let W = { A c Y : f-1(A) :- M }. Suppose that M is a s-algebra. What can we conclude about W ? |
| W is a s-algebra in Y page 89 in 2nd measure |
| Let f : X -> Y. Let M c P(X). Let W = { A c Y : f-1(A) :- M }. Suppose that M is an ideal. Does W have to be an ideal, too? |
| YES. page 89 in 2nd measure |