Classify the following three functions as injective, surjective, bijective, or none of these:

Question

1. Let X = {a,b,c}, Y = {1,2,3} and Z = {P,Q,R,S}. Classify the following three functions as injective, surjective, bijective, or none of these:

   · f:X→X given by f(a)=c, f(b)=a and f(c)=c
   · g:Y →X given by g(1)=b, g(2)=a and g(3)=c
   · h:X→Z given by h(a)=P, h(b)=S and h(c)=Q

Answer 1

Function f: X to itself omits b as an output (permissible as an injective function), but c has more than one input (cannot be injective). Cannot be surjective (but b is omitted). f is not a function—none.

Function g: the inputs and outputs form perfect pairs with nothing left out (the sets are the same size). Bijective.

Function h: R is omitted as an output and other mapping is unique—injective.