Suppose f : N→N satisestherecurrencerelation f(n + 1) (f(n) 2 if
f(n)iseven 3f(n)+ 1 if f(n)isodd . Notethatwiththeinitialcondition
f(0) 1,thevaluesofthefunction are: f(1) 4, f(2) 2, f(3) 1, f(4) 4,
and so on, the images cyclingthroughthosethreenumbers. Thus f
isNOTinjective(andalso certainlynotsurjective).
Mightitbeunderotherinitialconditions?3 (a) If f
satisestheinitialcondition f(0) 5,is f injective? Explain
whyorgiveaspecicexampleoftwoelementsfromthedomain withthesameimage.
(b) If f satisestheinitialcondition f(0) 3,is f injective? Explain
whyorgiveaspecicexampleoftwoelementsfromthedomain withthesameimage.
(c) If f satisestheinitialcondition f(0) 27,thenitturnsoutthat
f(105) 10 and no two numbers less than 105 have the same...