All Left Descent Subset Formulas Represent Disjoint Sets

Each stage of separation of the elements in a set which is a node in the abstract predecessor tree results in the formation of a leaf node set and two sets which are parental to nodes in the next generation.

Any given extension element is parental to a left descent which is completely determined.  Every left descent can be described by the set of formulas which characterize the contents of the sets which constitute it.  The formulas which describe the immediate children are always disjoint,  even in those cases where b and s steps result in the new c values which approach the same values(e.g. (4c-1)/3 from a b-step and (2(c+d)-1)/3 from an s-step when c is nearly as large as d).

Using the formulas provides proof that the subsets are disjoint.

Thus the three subsets involved in tree elaboration (parental,  and b and s children)  are disjoint with one another at any one level in the predecessor tree. The leaf subset is a subset of the parental set.  Its elements will never be subsetted in the way those parental to the b and s branches are.

On the way back up the tree after a leaf set has been identified,    subsets of the parental nodes at each level of the predecessor tree are developed.  That these are indeed subsets is quickly verified by application of the theorem relating the constants of the formulas for two subsets and the gcd of their coefficients.  This means that every leaf set identified can be mapped back into a subset within every one of its parental nodes back to and including the root node of extension elements.  We have also shown that every path descendent from any parental node terminates at length by reaching a leaf node.  This implies that every parental set will ultimately be completely assigned to subsets associated with particular left descents.

Subsequent generations of the abstract tree result in further subsetting of the sets established higher in the tree; once subsets are established as disjoint at any level of the predecessor tree,  there is no possibility that an overlap will develop at later generations.  This establishes that every left descent subset,  when fully characterized as to its root extension and its terminating leaf set,  is unique.

The view of the abstract predecessor tree as reflecting the results of a sieving process (either observed or from first principles) by mapping all paths in the abstract tree into initial  (i.e. with n=0)  left descents which each define a sieving assembly which,  along with its subsequent instantiations,  sieves all the odd positive integers.  This provides an indirect link between the abstract tree and the predecessor tree and indicates that both contain all the odd positive integers.  The subsets in the fully subsetted abstract predecessor tree correspond to the complete set of instantiations of any given element in a left descent. 

We also provide an algorithm which maps any odd positive integer into its unique place in the abstract predecessor tree.  This unique placement of every odd positive integer in the abstract tree also indicates that the integers in the predecessor tree are all unique.

My Collatz Home Page       Index to Terms Used