📝Set theory: functions

  • function: f:ABf : A \to B is a function iff xA.!yBf(x)=y\forall x \in A. \exists! y \in B \ni f(x) = y (i.e., it must be total)

    • AA is domain
    • BB is codomain
    • Image of function ff is a set of possible values (y’s)

      • yImage(f)    xAf(x)=yy \in Image(f) \iff \exists x \in A \ni f(x) = y
  • injective function: f:ABf : A \to B is injective iff x1,x2A.f(x)=f(y)    x=y\forall x_1, x_2 \in A. f(x) = f(y) \implies x = y

    • no two x’s map to the same y
  • surjective function: f:ABf : A \to B is surjective iff yB.xAf(x)=y\forall y \in B. \exists x \in A \ni f(x) = y

    • for all y, there is an x
  • bijective functions: is a function that is both injective and surjective

    • bijective functions have an inverse