Find all non-empty finite sets of integers A and B with the following properties:
(i) Whenever x∈A, x+1∈B.
(ii) Whenever x∈B, x2-4∈A.
(In reply to
Semi-analytical solution (spolier) by Steve Herman)
You were on the right track. I did a similar thing except for x in A.
Going from your x in B, try any x>=3. Then its successor is greater than x, which then generates an infinite sequence of integers for repeated applications. However we are asked for finite sets, therefore x<3 for any x in B.
Do the same thing but this time for x<-3. Then the immediate successor is a positive integer greater than 3, which by the earlier observation generates an infinite sequence of integers. Therefore x>-3 for any x in B.
Combine these two restrictions and we get the elements of a valid set B must come from {-2, -1, 0, 1, 2}. Then it is simple to do by hand (no computer needed) to generate the sets that Larry bruteforced.
re: Semi-analytical solution (spolier)
