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Classification of injective mappings and number sequences
ALEXANDER M. SUKHOTIN, TATIJANA A. SUKHOTINA
Department of Higher Mathematics
Tomsk Polytechnic University
30, Aven. Lenin, Tomsk, 634050
Russia
Abstract: New concepts are entered in the theory of injective mappings and in the theory of numerical sequences such as: a precise pair of variables, a divergent sequence, a convergent in itself sequence etc. The new methodological approach has allowed to classify injective mappings and numerical sequences and to prove some paradoxical from the classical point of view the statements on the analysis: the existence of infinity large Cauchy’s sequences has made possible and necessary the introduction of infinity large numbers.
Key-Words: Exact surjectivity, Antisurjectivity, Number Sequence, Divergent Sequence, Infinitely Large Numbers.
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1 Introduction
The Great scientist of XVII century G. Galilei, having discovered that the quantities of natural numbers and their quadrates are equal, has bequeathed to the successors to be very cautious at an operation with infinite amounts: "… the properties of equality, and also greater and smaller values have no the places there, where the matter goes about the infinity, and they can be applied only to finite amounts" [1, p. 140-146]. The ignoring of this warning has entered into the mathematical folklore some false hypotheses together with its proofs that contain incorrect reasoning. These and contiguous by them problems were as a subject of learning in this work. The new procedure enabled us to overcome the above difficulties.
The injective mappings j: and the properties of numerical sequences have been analyzed in this work. Classification of the investigated objects has become one of the results of our research. This theory is borne out by the facts too.
2 Problems Formulation
For finite sets A and B the check of mapping surjectivity does not cause difficulties. On the contrary, the similar procedure for mappings of infinite sets is not such trivial. Injective mappings j: , except for obvious antisurjective such, as , are considered bijective by default in the traditional mathematical texts. The proof of surjectivity criteria for injective mappings j: and their classification made up the first problem, which was solved in this paper. The second problem – the research of properties of numerical sequences and their classification has been solved due to the introduction of positive definition of a divergent numerical sequence. The main result of these researches has been formulated in the following form:
Theorem 1. Any fundamental number sequence (а) satisfies to the following condition:
(1)
or, that is the same,
. (2).
3 Solutions of problems
3.1 About properties of injective mappings N®N
The infinity of set of natural numbers is understood in connection with a principle of a mathematical induction as unbounded possibility of transition from (n) to (n+1). More common phrases "at a passage to the limit in F(n)" and "at in F(n)" mean the following:
. (3)
The principle of the passage to the limit (3) and an uniform ordering of set of natural variables make possible the introduction of the following concept.
Definition°1. The pair (n, m) of variables, n, mÎN, is named as a precise pair at , if
. (4)
Let there is an infinite sequence of natural numbers x=() and let. The sequence x (the splitting (x)) divides set N into the segments , , . The injective mapping j: and splitting (x) induce three sequences: , and of the non-negative integers under the following formulas:
, , , (5)
, (6)
,
. (7).
From the condition (5) follows that . Figure 1 illustrates the mapping , where .
Fig. 1
Here , , .
From determining conditions (5)(7) follows, that , =. It is easy to prove the following
Statement 1. Sequences and , determined in any pair (x, j), satisfy for almost to one and only to one of three following conditions:
1) Û,
2) Û, (8)
3) $ÎN: Û.
The following below statement is a consequence of conditions (8):
Statement 2. If such splitting (x) of sets N on pieces and number exist for an injection, that for " , then for any number С>0 new splitting () of set N can be received by means of corresponding enlargement of pieces, that inequalities and С<inf{, } will be fair for pieces of splitting ().
Classification of injective mappings is carried out to following two properties: 1) a triviality, not trivial boundedness, unboundedness of the sequence , 2) fullfilment or default of a condition
Conditions (5)(7) allow to separate all injective mappings on three not crossed classes:
Definition 2. Injective mapping is determined as exact ones, if there is a sequence x=() such, that " =0.
Definition 3. Injective mapping is determined as potentially surjective ones, if there are a sequence x=() and number >0 such, that almost " 0<£ and besides .
Definition 4 Injective mapping is determined as antisurjective ones, if at least one of two following conditions is carried out:
1) "С0, "x=() $ : С,
2) "x=() ${С(j),°,°}: " n, (9)
n>ÎN: =С(j).
The necessary criterion exact or potential surjectivity of injections is formulated on the basis of the classification given above as follows:
Theorem 2. Injective mapping is exact or potentially surjective ones in only case when the following below two conditions have been satisfied:
1) $x=(), , $0:
" (10)
2) .
Theorem 3. For injective mappings of first two classes the following below limiting equality are fair:
a) lim, (11)
b) lim, if this limit exists. (12)
● a) There is always for exact surjective injective mappings and, hence, the conditions (11) are fair for such mappings. Generally, the opportunity of construction of splitting () of set N for the given injection , which provides the existence of corresponding limit (11), follows out of Statement 2.
b) We shall assume opposite lim, 1<k¥. Then there is a number such, that n for almost all n and the any a, . Hence, , that means the unboundedness of sequence , that contradicts to the condition of the Theorem 2. The similar conclusion can be received for function , if the k satisfies to the inequality 0<k<1 of (12).■
The statement following below is consequence of (10) for infinite subsets of set N (comp. [2, p. 20]):
Theorem 4. A bijection does not exist between set N of natural numbers and its own subset АÌN.
Theorem 4 can be proved with the help of the following below Theorem 5 (see [3]).
Theorem 5. Let A and B be own subsets of set N also there is an injection . Then this injection can be continued up to bijection .
The opportunity of such continuation is proved by means of consecutive designing of points of set on a diagonal of set and on the diagram of function (see also Example 4 lower).
Theorem 6. The equivalence (®0)Û(®0) is fair for any number series (a).
l а) The implication ((®0)Þ(®0) follows from . b) Further, let ®0, let both and be the partial sums of a number series (a). If , then the remainder of series (a) is determined by equality . Therefore,
. (13)
Let formula determine an injection . Then it is possible to consider by virtue the first of conditions (10) of Theorem 2 that in the limit condition (13). Therefore, it is possible to consider a pair in (13) as a precise pair (see (4)), therefore, =0, because ®0. n
3.2 Examples
Example 1 (G..Galileja's paradox [1]). It is obvious, that a mapping defined by the formula , satisfies to both conditions (9). Such mappings can be named as total antisurjective ones.
Example 2 (to Th. 3). Let for some splitting (x) of set N on pieces a mapping be the change of the order on the pieces , on opposite ones. Then this mapping belongs to the first class since . If splitting (x) is limited ones, i.e. , then lim (see (12)). If splitting (x) is not limited, then does not exist, and lim (see (11)).
Example 3 (to Th. 6) It is know harmonic series is defined by formula: . For this series (see [4, p. 319]) ==lnm++, ==lnk++, where is Euler’s constant and ®0.
Therefore, .
Hence, ==0 for harmonic series.
Example 4. (to Th.5). Let , and let be an injection which is defined by the formula: . If and then . Further, let A=(2, 3, 4, …, m, …), (3, 2, 4, …, m, …), …, (3, 4, 5, …, k+1, 2, k+2, k+3, …), , be the simply ordered subsets of the set N and let the bijection be the transform of a pair (2, n+2) into ones (n+2, 2) and , .
Now we shall determine a bijection recurrently by means the rules: ,
, , i.e. , ËB.
By virtue of (3), we determine a bijection by means the formula: . At last, we determine a bijection by following way:
It is easy to see what , i.e. the restriction of the bijection on each subset is a bijection between sets and . Therefore, the bijection is a continuation of the injection . Besides, =B, because 2ÏB.
3.3 Properties and classification of number sequences
A sequence (а), (а)()(), where nÎN, , is named a convergent to number sequence and there are speaking in this case that this sequence converges to number a, if a following below condition (14) is fulfilled [5, S. 46], [6, S. 63]:
. (14)
The number sequence (а) is named a fundamental sequence (regular sequence, Cauchy's sequence (CS)) (see [7, p. 355], comp. with [6, S. 81, 85-86]), if
. (15)
It is easy to prove that the limiting equality (15) is equivalent “on the language (e, n(e))” to following condition:
(16)
It is possible to consider a pair (m, n) of variables n and m in (15) as a precise ones (see (4)): . The validity of Theorem 1 follows from both the (15) and the (4) at p=1.
Definition 5. A number sequence (а) is named "convergent in itself" ones (CIS), if (2) is fair or, that is same, if
(17)
The term "CIS" has been used in [5, S. 51] as a synonym for "CS". It is obvious, that the convergent to number a sequence (а) will be and CIS too. The inverse statement has no a place generally.
Theorem 7. The following below limit equality is fulfilled for a convergent (or convergent in itself) number sequence at any natural number p
. (18)
lThe validity of Theorem 7 follows from conditions (14), (or from (17), accordingly) and from the following equality:
n(19)
Definition 6. The number sequence (a) is termed a divergent ones (DS), if there are two infinite subsequences such, that the following condition
(20)
is fulfilled (comp. with [5, p. 46]).
The following statement is fair by virtue of conditions (15), (16) and (20):
Theorem 8. Any numerical sequence is either regular sequence, or divergent sequence, i. е.
"(a) (a)Î{CS}È{DS}, {CS}Ç{DS}=Æ, (21)
The direct proof of Theorem 1 is stated below.
l We suppose the opposite judgment that there is a sequence convergent in itself but ones is not being fundamental. By virtue (21) this sequence will be divergent, that means the condition (20) is fair:
(20¢)
By virtue of infinity of sequences there are as is wished many pairs , among all pairs (m, k)Î() such, that for some and, for example, . Then by virtue (20¢)
(22)
Let at (22) then by virtue of conditions (17) - (19), which are fair for any СIS, we have
(23)
then an inconsistency of inequalities (22) and (23) proves the theorem.
Let , pÎN in (21). In this case by virtue of equality (18), which is fair for any CIS, we have
. (24)
Let . Any pair () is not exact ones (see (4)) but by virtue of that the accepts all values , n, there is, at least, one pair from all pairs (), such, that . Then the required contradiction follows out from the inequalities (20¢) and (24):
.n
A sequence (a) is termed as limited ones, if $С>0 C. If this condition is not carried out, the number sequence (а) is termed as unlimited.
Example 5. Let the sequence (s) determine for all by the formula: , then it is limited and divergent ones. We shall prove below that the sequence (s) hasn’t any convergent subsequence.
lWe shall assume opposite that (, , ) and . Let .
By virtue of we have
,
or . This limiting equality will take place, if at least one of two limiting equalities
,
is fair. The last conditions are equivalent to corresponding limiting equalities at relevant integer numbers and:
,
(25)