2000 MSC primary 00A05; sec. 00A71, 03E10, 28B15, 40C99
Uniquantities (Essential)
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
RUAG Aerospace Services GmbH, Germany
Mathematical Journal
of the "Collegium" All World Academy of Sciences
Munich (Germany)
5(2005),3
The cardinality is sensitive to finite unions of disjoint finite sets only. Each measure is finitely sensitive within a certain dimension. They give continuum cardinality cand (0 or +∞), respectively, for clearly distinct point sets between two parallel lines or planes differently distant from one another. The measures (probabilities) cannot discriminate the empty set ∅ and null sets (impossible and differently possible events, respectively). For uncountable sets of elementary events, also uncountable sums of their probabilities should be considered.
Definition 1. The uniquantity of a quantiset is the non-positional and possibly uncountable quantisum of the own (inside) quantities of all elements (bases) of the quantiset
Q(A) = Q{... , qa , ... , rb , ... , sc , ...}° = ...+ q + ...+ r + ... + s + ...
extending a point (zero-dimensional) measure (number of elements) bysatisfying the following axioms:
(A1) a uniquantity is completely algebraically additive, commutative, and associative, say Q(... ∪° A \° ... \° B ∪° ... ∪° C \° ... \° D ∪° ...) = ...+ Q(A) - ... - Q(B) + ... + Q(C) - ... - Q(D) + ...;
(A2) quantifying a quantiset implies multiplying its uniquantity: Q(tA) = tQ(A) ;
(A3) uniquantities of Cartesian quantisets products are their uniquantities products: Q(... × A × ... × B × ...) = ... × Q(A) × ... × Q(B) × ...;
(A4) norming any bases in a quantiset does not affect its uniquantity, say Q{... , qa , ..., rb , ...}°= Q{... , q||a||, ... ,rb ,...}°;
(A5) if all distances between the bases of a quantiset are bounded above in common and its mapping preserves both the distances (is an isometry) and the own quantities of the bases in the quantiset, the mapping preserves its uniquantity;
(A6) uniquantity determination (as a set quantioperation) commutes with a passage to the limit,
(A7) the uniquantities of some chosen canonical sets coincide with their cardinalities, in particular: Q{a} = 1 for any object a ; Q(Z) = 2ℵ (in place of ℵ0 for simplicity of notation), or, equivalently, Q(D) = ℵ for D =° {1/20, 1, 2, 3, 4, ...}°; Q|0, 1| = c.
Remark 2. For disjoint sets, the cardinality and each measure are only countably additive. Because of set absorption, they are not algebraically additive and even nonadditive. The quantisets uniquantities are perfectly sensitive (there is no absorption).
Corollary 3. The uniquantity of the empty set (a finite ordinary set) is zero (the number of its elements, respectively).
Corollary 4. For the sets of all natural numbers, positive and negative integers, Q(N) = ℵ + 1/2, Q(Z+) = ℵ - 1/2, Q(Z-) = ℵ - 1/2.
Corollary 5. Adding any integer to each number in N or Z+ (Z-) implies subtracting (adding) this integer from (to, respectively) the uniquantity, say Q{z , z+1, z+2, ...} = Q(N) - z .
Definition 6. Q{a + bn | n ∈ D} = ℵ/|b| - a/b (a , b ∈ R°; b ≠ 0).
Corollary 7. Q{a + bn | n ∈ N} = ℵ/|b| - a/b + 1/2 (a , 0 ≠ b ∈ R°);
Q{z , z + 1, z + 2, ...} = Q{z + 1 × n | n ∈ N} = ℵ - z + 1/2;
Q{2n | n ∈ N} = ℵ/2 + 1/2; Q{1 + 2n | n ∈ N} = ℵ/2.
Definition 8. If a real function f(N) has inverse f-1, then Q{f(n) | n ∈ D} is (f-1(À))0 where ( )0 is a correction such that
Q{- f(n) | n ∈ D} = Q{f(n) | n ∈ D}, e.g.
Q{a + bnk | n ∈ D} = (ℵ/|b| - a/b)1/k ;
Q{a + bcn | n ∈ D} = logc(ℵ/|b| - a/b).
Remark 9. The existence of f-1 is not necessary for Q{f(n) | n ∈ D} to be defined. For example (a , b , c ∈ R°; c ≠ 0),
Q{a + bn + cn2 | n ∈ D} = [ℵ/|c| - a/c + b2/(2c)2]1/2 - b/(2c)
giving Definition 6 as c → 0, the correcting sign of modulus jumping from c to b with b ≠ 0 instead of c ≠ 0.
Introduced uniquantities apply to information problems etc.