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28-502: In 2022, the cargo traffic in the seaport equaled 1,212,000 tons, comprising 1% of all cargo traffic in Polish seaports and the port was entered by 227 ships with gross tonnage of more than 100. In January 2019 Port of Police and PKP have signed an agreement to create a direct railway link with the port to improve the delivery of the cargo. A direct track with accompanying infrastructure will be constructed from Police railway station to

56-634: A binary relation pairing elements of set X with elements of set Y to be a bijection, four properties must hold: Satisfying properties (1) and (2) means that a pairing is a function with domain X . It is more common to see properties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y . Functions which satisfy property (3) are said to be " onto Y " and are called surjections (or surjective functions ). Functions which satisfy property (4) are said to be " one-to-one functions " and are called injections (or injective functions ). With this terminology,

84-621: A bijection between them. More generally, two sets are said to have the same cardinal number if there exists a bijection between them. A bijective function from a set to itself is also called a permutation , and the set of all permutations of a set forms its symmetric group . Some bijections with further properties have received specific names, which include automorphisms , isomorphisms , homeomorphisms , diffeomorphisms , permutation groups , and most geometric transformations . Galois correspondences are bijections between sets of mathematical objects of apparently very different nature. For

112-417: A bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto". Consider the batting line-up of a baseball or cricket team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). The set X will be the players on the team (of size nine in the case of baseball) and

140-405: A function f : X → Y is bijective if and only if it satisfies the condition Continuing with the baseball batting line-up example, the function that is being defined takes as input the name of one of the players and outputs the position of that player in the batting order. Since this function is a bijection, it has an inverse function which takes as input a position in the batting order and outputs

168-453: A function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain Y . Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is, the inverse function exists and is also a bijection. Functions that have inverse functions are said to be invertible . A function is invertible if and only if it is a bijection. Stated in concise mathematical notation,

196-497: A mathematical formula. Gross tonnage is based on "the moulded volume of all enclosed spaces of the ship" whereas net tonnage is based on "the moulded volume of all cargo spaces of the ship". In addition, a ship's net tonnage is constrained to be no less than 30% of her gross tonnage. The gross tonnage calculation is defined in Regulation 3 of Annex 1 of The International Convention on Tonnage Measurement of Ships, 1969 . It

224-459: A ship's manning regulations, safety rules, registration fees, and port dues, whereas the older gross register tonnage is a measure of the volume of only certain enclosed spaces. The International Convention on Tonnage Measurement of Ships, 1969 was adopted by IMO in 1969. The Convention mandated a transition from the former measurements of gross register tonnage (grt) and net register tonnage (nrt) to gross tonnage (GT) and net tonnage (NT). It

252-447: Is a function such that each element of the second set (the codomain ) is the image of exactly one element of the first set (the domain ). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if and only if it is invertible ; that is, a function f : X → Y {\displaystyle f:X\to Y}

280-472: Is any relation R (which turns out to be a partial function) with the property that R is the graph of a bijection f : A′ → B′ , where A′ is a subset of A and B′ is a subset of B . When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation . An example is the Möbius transformation simply defined on the complex plane, rather than its completion to

308-408: Is based on two variables, and is ultimately an increasing one-to-one function of ship volume: The value of the multiplier K increases logarithmically with the ship's total volume (in cubic metres) and is applied as an amplification factor in determining the gross tonnage value. K is calculated with a formula which uses the common or base-10 logarithm : Once V and K are known, gross tonnage

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336-675: Is bijective if and only if there is a function g : Y → X , {\displaystyle g:Y\to X,} the inverse of f , such that each of the two ways for composing the two functions produces an identity function : g ( f ( x ) ) = x {\displaystyle g(f(x))=x} for each x {\displaystyle x} in X {\displaystyle X} and f ( g ( y ) ) = y {\displaystyle f(g(y))=y} for each y {\displaystyle y} in Y . {\displaystyle Y.} For example,

364-489: Is calculated using the formula, whereby GT is a function of V: which by substitution is: Thus, gross tonnage exhibits linearithmic growth with volume, increasing faster at larger volumes. The units of gross tonnage, which involve both cubic metres and log-metres, have no physical significance, but were rather chosen for historical convenience. Since gross tonnage is a bijective function of ship volume, it has an inverse function , namely ship volume from gross tonnage, but

392-402: Is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function , i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the symmetric inverse semigroup . Another way of defining the same notion is to say that a partial bijection from A to B

420-430: The multiplication by two defines a bijection from the integers to the even numbers , which has the division by two as its inverse function. A function is bijective if and only if it is both injective (or one-to-one )—meaning that each element in the codomain is mapped from at most one element of the domain—and surjective (or onto )—meaning that each element of the codomain is mapped from at least one element of

448-413: The category Grp of groups , the morphisms must be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphisms which are bijective homomorphisms. The notion of one-to-one correspondence generalizes to partial functions , where they are called partial bijections , although partial bijections are only required to be injective. The reason for this relaxation

476-404: The composition g ∘ f {\displaystyle g\,\circ \,f} of two functions is bijective, it only follows that f is injective and g is surjective . If X and Y are finite sets , then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Indeed, in axiomatic set theory , this is taken as

504-419: The definition of "same number of elements" ( equinumerosity ), and generalising this definition to infinite sets leads to the concept of cardinal number , a way to distinguish the various sizes of infinite sets. Bijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are not always the isomorphisms for more complex categories. For example, in

532-432: The domain. The term one-to-one correspondence must not be confused with one-to-one function , which means injective but not necessarily surjective. The elementary operation of counting establishes a bijection from some finite set to the first natural numbers (1, 2, 3, ...) , up to the number of elements in the counted set. It results that two finite sets have the same number of elements if and only if there exists

560-586: The inverse cannot be expressed in terms of elementary functions . A root-finding algorithm may be used for obtaining an approximation to a ship's volume given its gross tonnage. The formula for exact conversion of gross tonnage to volume is: where ln {\displaystyle \ln } is the natural logarithm and W {\displaystyle W} is the Lambert W function . Bijective A bijection , bijective function , or one-to-one correspondence between two mathematical sets

588-598: The player who will be batting in that position. The composition g ∘ f {\displaystyle g\,\circ \,f} of two bijections f : X → Y and g : Y → Z is a bijection, whose inverse is given by g ∘ f {\displaystyle g\,\circ \,f} is ( g ∘ f ) − 1 = ( f − 1 ) ∘ ( g − 1 ) {\displaystyle (g\,\circ \,f)^{-1}\;=\;(f^{-1})\,\circ \,(g^{-1})} . Conversely, if

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616-425: The same position in the list. In a classroom there are a certain number of seats. A group of students enter the room and the instructor asks them to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. What the instructor observed in order to reach this conclusion

644-732: The seaport itself. The Port of Police has access to the Baltic Sea through the Szczecin Lagoon , Świna strait. This West Pomeranian Voivodeship location article is a stub . You can help Misplaced Pages by expanding it . Gross tonnage Gross tonnage ( GT , G.T. or gt ) is a nonlinear measure of a ship's overall internal volume. Gross tonnage is different from gross register tonnage . Neither gross tonnage nor gross register tonnage should be confused with measures of mass or weight such as deadweight tonnage or displacement . Gross tonnage, along with net tonnage ,

672-479: The set Y will be the positions in the batting order (1st, 2nd, 3rd, etc.) The "pairing" is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in

700-586: Was defined by the International Convention on Tonnage Measurement of Ships, 1969 , adopted by the International Maritime Organization (IMO) in 1969, and came into force on 18 July 1982. These two measurements replaced gross register tonnage (GRT) and net register tonnage (NRT). Gross tonnage is calculated based on "the moulded volume of all enclosed spaces of the ship" and is used to determine things such as

728-451: Was provided to allow ships time to adjust economically, since tonnage is the basis for satisfying manning regulations and safety rules. Tonnage is also the basis for calculating registration fees and port dues. One of the convention's goals was to ensure that the new calculated tonnages "did not differ too greatly" from the traditional gross and net register tonnages. Both GT and NT are obtained by measuring ship's volume and then applying

756-428: Was that: The instructor was able to conclude that there were just as many seats as there were students, without having to count either set. A bijection f with domain X (indicated by f : X → Y in functional notation ) also defines a converse relation starting in Y and going to X (by turning the arrows around). The process of "turning the arrows around" for an arbitrary function does not, in general , yield

784-607: Was the first successful attempt to introduce a universal tonnage measurement system. Various methods were previously used to calculate merchant ship tonnage, but they differed significantly and one single international system was needed. Previous methods traced back to George Moorsom of Great Britain 's Board of Trade who devised one such method in 1854. The tonnage determination rules apply to all ships built on or after 18 July 1982. Ships built before that date were given 12 years to migrate from their existing gross register tonnage (GRT) to use of GT and NT. The phase-in period

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