properties of a function

Let X,Y be sets and f:XY be a function. For any AX, define


and any BY, define


So f(A) is a subset of Y and f-1(B) is a subset of X.

Let A,A1,A2,Ai be arbitrary subsets of X and B,B1,B2,Bj be arbitrary subsets of Y, where i belongs to the index setMathworldPlanetmathPlanetmath I and j to the index set J.  We have the following properties:

  1. 1.

    If A1A2, then f(A1)f(A2). In particular, f(A)f(X).

  2. 2.

    f(A1A2)=f(A1)f(A2). More generally, f(iAi)=if(Ai).

  3. 3.

    f(A1A2)f(A1)f(A2). The equality fails in the example where f is a real function defined by f(x)=x2 and A1={1}, A2={-1}. Equality occurs iff f is one-to-one:

    Suppose f(x)=f(y)=z. Pick A1={x} and A2={y}. Then f(A1A2)=f(A1)f(A2)={z}. This means that A1A2. Since both A1 and A2 are singletons, A1=A2, or x=y.

    Conversely, let’s show that f is one-to-one then f(A1A2)=f(A1)f(A2). To do this, we only need to show the right hand side is included in the left, and this follows since if xf(A1)f(A2) then for some a1A1 and a2A2 we have x=f(a1)=f(a2). As f is one-to-one, a1=a2 and so a1 lies in A1A2 and x is in f(A1A2).

    More generally, f(iAi)if(Ai).

  4. 4.

    f(A1)-f(A2)f(A1-A2): If yf(A1)-f(A2), then y=f(x) for some xA1. If xA2, then y=f(x)f(A2) as well, a contradictionMathworldPlanetmathPlanetmath. So xA1-A2, and y=f(x)f(A1-A2). The inequality is strict in the case when f: given by f(x)=1, and A1= and A2={2}.

  5. 5.

    Af-1f(A). Again, one finds that equality fails for the real function f(x)=x2 by selecting A={1}. Equality again holds iff f is injective:

    Suppose xf-1f(A). By definition this means that f(x)=f(a) for some xA, and since f is injective we have x=aA. It follows that f-1f(A)A. Convserly, if f(x)=f(y)=z, then {x,y}=f-1f({x,y})=f-1({z}). On the other hand {x}=f-1f({x})=f-1({z}). So {x,y}={x}, x=y.

  6. 6.

    If B1B2, then f-1(B1)f-1(B2). In particular, f-1(B)f-1(Y).

  7. 7.

    f-1(B1B2)=f-1(B1)f-1(B2). More generally, f-1(jBj)=jf-1(Bj).

  8. 8.

    f-1(B1B2)=f-1(B1)f-1(B2). More generally, f-1(jBj)=jf-1(Bj).

  9. 9.

    f-1(Y-B)=X-f-1(B). As a result, f-1(B1-B2)=f-1(B1)-f-1(B2).

  10. 10.

    ff-1(B)B. Yet again, one finds that equality fails for the real function f(x)=x2 by selecting B=[-1,1]. Equality holds iff f is surjectivePlanetmathPlanetmath:

    Suppose f is onto. Pick any yBY. Then y=f(x) for some xX. In other words, xf-1(B) and hence y=f(x)ff-1(B). Now suppose the convserse, then pick B=Y, and we have Y=ff-1(Y)=f(X).

  11. 11.

    Combining 10 and 5, we have that ff-1f(A)=f(A) and f-1ff-1(B)=f-1(B). Let’s show the first equality:

    From 5, Af-1f(A), so that f(A)ff-1f(A) (by 1). Set B=f(A). Then by 10, ff-1f(A)=ff-1(B)B=f(A).


  • f-1f and ff-1 the compositionsMathworldPlanetmath of the function and its inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath as defined at the beginning of the entry, so that f-1f(A)=f-1(f(A)) and ff-1(B)=f(f-1(B)).

  • From the definition above, we see that a function f:XY induces two functions [f] and [f-1] defined by

    [f]:2X2Y such that [f](A):=f(A) and
    [f-1]:2Y2X such that [f-1](B):=f-1(B).

    The last property 11 says that [f] and [f-1] are quasi-inverses of each other.

  • f is a bijection iff [f] and [f-1] are inverses of one another.

Title properties of a function
Canonical name PropertiesOfAFunction
Date of creation 2013-03-22 16:21:38
Last modified on 2013-03-22 16:21:38
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 24
Author CWoo (3771)
Entry type Definition
Classification msc 03E20
Related topic PropertiesOfFunctions