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Bijection - Composition and inverses

Bijection - Composition and inverses: Encyclopedia II - Bijection - Composition and inverses

A function f is bijective if and only if its inverse relation f-1 is a function. In that case, f-1 is a bijection. The composition (mathematics) gf of two bijections f XY and g YZ is a bijection. The inverse of gf is (gf)-1 = (f-1)(g-1). On the other hand, if the composition g o f of two functions is bijective, we can only say ...

Bijection, Bijection - Bijections and cardinality, Bijection - Bijections and category theory, Bijection - Category theory, Bijection - Composition and inverses, Bijection - Examples and counterexamples, Bijection - Properties, injective function, isomorphism, permutation, symmetric group, surjective function

Bijection: Encyclopedia II - Bijection - Composition and inverses

Bijection - Composition and inverses

A function f is bijective if and only if its inverse relation f^-1 is a function. In that case, f^-1 is a bijection.

The composition (mathematics) gf of two bijections f XY and g YZ is a bijection. The inverse of gf is (gf)^-1 = (f^-1)(g^-1).

On the other hand, if the composition g ^o f of two functions is bijective, we can only say that f is injective and g is surjective.

A relation f from X to Y is a bijective function if and only if there exists another relation g from Y to X such that gf is the identity function on X, and fg is the identity function on Y.

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Set, Set theory, Wikipedia cleanup, axiomatic set theory, cardinal number, category, composition (mathematics), diffeomorphism, epimorphisms, exponential function, factorial, finite, function, group, homeomorphism, identity function, if and only if, infinite, injective, injective function, integers, inverse relation, isomorphism, isomorphisms, mathematics, monomorphisms, natural logarithm, permutation, permutation group, projective map, real line, set, sets, surjective, surjective function, symmetric group, up to isomorphism

Adapted from the Wikipedia article "Composition and inverses", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki

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