properties of functions

Let f:XY be a function. Let (Ai)iI be a family of subsets of X, and let (Bj)jJ be a family of subsets of Y, where I and J are non-empty index setsMathworldPlanetmathPlanetmath.

Then, it is easy to prove, directly from definitions, that the following hold:

  • f(iIAi)=iIf(Ai) (i.e., the image of a union is the union of the images)

  • f(iIAi)iIf(Ai) (i.e., the image of an intersectionMathworldPlanetmathPlanetmath is contained in the intersection of the images)

  • Af-1(f(A)) for any AX (where f-1(f(A)) is the inverse imagePlanetmathPlanetmath of f(A))

  • f(f-1(B))B for any BY

  • f-1(YB)=Xf-1(B) for any BY

  • f-1(jJBj)=jJf-1(Bj) (the inverse image of a union is the union of the inverse images)

  • f-1(jJBj)=jJf-1(Bj) (the inverse image of an intersection is the intersection of the inverse images)

  • f(f-1(B))=B for every BY if and only if f is surjectivePlanetmathPlanetmath.

For more properties related specifically to inverse images, see the inverse image ( entry.

Further, the following conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (for more, see the entry on injective functions):

  • f is injective

  • f(ST)=f(S)f(T) for all S,TX

  • f-1(f(S))=S for all SX

  • f(S)f(T)= for all S,TX such that ST=

  • f(ST)=f(S)f(T) for all S,TX

Title properties of functionsPlanetmathPlanetmath
Canonical name PropertiesOfFunctions
Date of creation 2013-03-22 14:59:54
Last modified on 2013-03-22 14:59:54
Owner yark (2760)
Last modified by yark (2760)
Numerical id 20
Author yark (2760)
Entry type Result
Classification msc 03E20
Related topic PropertiesOfAFunction