SUMMARY / RELATED TOPICS

Józef Walczak

Józef Walczak was a football player and manager who played for ŁKS Łódź during his playing career making two international appearances for Poland between 1954-1956, went on to manage 8 different teams. Walczak started his playing career with his local club ŁKS Łódź playing with ŁKS for two seasons, winning promotion in the first season before finishing second in the top division the season after, he moved to Zawisza Bydgoszcz for two seasons. He returned to ŁKS 1957, he played for ŁKS during their golden years helping them to their first I liga title in 1958, won the Polish Cup with ŁKS in 1957 the only time the team have won the competition. In total for ŁKS Łódź, Walczak made a total of 211 games scoring 11 goals. After his playing career Walczak went on to manage ŁKS Łódź, Włókniarz Łódź, Włókniarz Pabianice, Motor Lublin, Bałtyk Gdynia, Lechia Gdańsk, Stal Mielec and Cracovia, his management career saw him finishing second in the II liga with Lechia in 1978, while achieving a third-place finish in the top division with Stal Mielec in 1982.

ŁKS Łódź I liga Winners: 1958 Runners-up: 1954 Third place: 1957 Polish Cup Winners: 1957 II liga Runners-up: 1953Zawisza Bydgoszcz II liga Third place: 1955 Lechia Gdańsk II liga Runners-up: 1977-78Stal Mielec I liga Third place: 1981-82

Bijection, injection and surjection

In mathematics, injections and bijections are classes of functions distinguished by the manner in which arguments and images are related or mapped to each other. A function maps elements from its domain to elements in its codomain. Given a function f: X → Y: The function is injective, or one-to-one, if each element of the codomain is mapped to by at most one element of the domain, or equivalently, if distinct elements of the domain map to distinct elements in the codomain. An injective function is called an injection. Notationally: ∀ x, x ′ ∈ X, f = f ⇒ x = x ′. Or, equivalently, ∀ x, x ′ ∈ X, x ≠ x ′ ⇒ f ≠ f; the function is surjective, or onto, if each element of the codomain is mapped to by at least one element of the domain. That is, the image and the codomain of the function are equal. A surjective function is a surjection. Notationally: ∀ y ∈ Y, ∃ x ∈ X such that y = f; the function is bijective if each element of the codomain is mapped to by one element of the domain. That is, the function is both injective and surjective.

A bijective function is called a bijection. An injective function need not be surjective, a surjective function need not be injective; the four possible combinations of injective and surjective features are illustrated in the adjacent diagrams. A function is injective if each possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective. An injective function is an injection; the formal definition is the following. The function f: X → Y is injective, if for all x, x ′ ∈ X, f = f ⇒ x = x ′; the following are some facts related to injections: A function f: X → Y is injective if and only if X is empty or f is left-invertible. Since every function is surjective when its codomain is restricted to its image, every injection induces a bijection onto its image. More every injection f: X → Y can be factored as a bijection followed by an inclusion as follows. Let fR: X → f be f with codomain restricted to its image, let i: f → Y be the inclusion map from f into Y.

F = i o fR. A dual factorisation is given for surjections below; the composition of two injections is again an injection, but if g o f is injective it can only be concluded that f is injective. Every embedding is injective. A function is surjective. In other words, each element in the codomain has non-empty preimage. Equivalently, a function is surjective. A surjective function is a surjection; the formal definition is the following. The function f: X → Y is surjective, if for all y ∈ Y, there is x ∈ X such that f = y; the following are some facts related to surjections: A function f: X → Y is surjective if and only if it is right-invertible, that is, if and only if there is a function g: Y → X such that f o g = identity function on Y. By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on a quotient of its domain. More every surjection f: X → Y can be factored as a non-bijection followed by a bijection as follows. Let X/~ be the equivalence classes of X under the following equivalence relation: x ~ y if and only if f = f.

Equivalently, X/~ is the set of all preimages under f. Let P: X → X/~ be the projection map which sends each x in X to its equivalence class ~, let fP: X/~ → Y be the well-defined function given by fP = f. F = fP o P. A dual factorisation is given for injections above; the composition of two surjections is again a surjection, but if g o f is surjective it can only be concluded that g is surjective. A function is bijective if it is both surjective. A bijective function is a bijection (one-to-one correspondenc