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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 A,B,C,D,E,F be non-empty sets. Suppose that (AxC) u (BxD) = ExF. Suppose that A n B = O. What can we conclude? |
| C = D Take any d:-D. Choose some b:-B. Now (b,d):-BxD. Hence (b,d):- ExF. Choose some (a,c):-AxC. Now (a,c):-ExF. We conclude that (a,d):- ExF. Hence (a,d):-AxC or (a,d):-BxD. But a:-A and AnB=O, h |
| Let A,B,C,D be non-empty sets. Prove that if A=B or C=D, then (A x C) u (B x D) = (A u B) x (C u D). |
| ___ |
| Let J1 be a semi-ring in X, and J2 a semi-ring in Y. Let u : J1 -> [0,oo] and y : J2 -> [0,oo] be additive. Let J = { E x F : E:-J1 and F:-J2 }. Let q : J -> |R*, q(E x F) = u(E) * y(F). What can we conclude about q? |
| q is additive page 89 in 1st measure |
| Let W be a ring of sets. Let y : W -> |R* be additive. Prove that for every A,B :- W y(A u B) + y(A n B) = ??? |
| y(A u B) + y(A n B) = y(A) + y(B) 1) A u B = A u (B\A) disjointly 2) y(A u B) = y(A) + y(B\A) 3) y(A u B) + y(A n B) = y(A) + y(B\A) + y(B n A) 4) y(A u B) + y(A n B) = y(A) + y(B) |
| Let W be a ring of sets. Let y : W -> |R* be additive. Prove that for every A,B :- W y(A) + y(B) = ??? |
| y(A) + y(B) = y(A u B) + y(A n B) PROOF: 0) A u B = A u (B\A) disjointly 1) y(A u B) + y(A n B) = y(A) + y(B\A) + y(B n A) 2) y(A u B) + y(A n B) = y(A) + y(B) |
| Let (T,>) be a directed set and let p:-T. Let P = { t:-T : t > p }. What can we say about (P,>) ? |
| (P,>) is also a directed set |
| Let W c P(X). Let A,B c X, x :- X. Suppose that x is W-adherent to A u B. What can we conclude? |
| x is W-adherent to A or x is W-adherent to B page 53 in gen top |
| Let W c P(X), A,B c X. Suppose that A,B are W-closed. What can we conclude? |
| 1) A u B is W-closed 2) A n B is W-closed page 54 in gen top |
| Let W c P(X), A,B c X. Suppose that A,B are W-open. What can we conclude? |
| A n B is W-open, page 55 in gen top A u B is W-open, page 56 in gen top |
| Let W c P(X). Let M c P(X) be a collection of W-closed sets. What can we conclude? |
| //\\M is W-closed page 55 in gen top |