Subset Relationship between Sets C and D

Let C be the set of all integers n such that n = 6r - 5 for some integer r. Let D be the set of all integers m such that m = 3s + 1 for some integer s. Prove or disprove: (a) C is a subset of D; (b) D is a subset of C.

In this problem, the focus is on understanding subset relationships between two sets defined through linear Diophantine equations. You're presented with two sets: C, described by numbers of the form 6r - 5, and D, described by numbers of the form 3s + 1. These represent integer sequences where each element is derived from linear expressions. A critical component of solving this problem involves recognizing how these sequences operate modulo some integer. The given forms suggest exploring these numbers modulo 3 and modulo 6 to predict their behavior and identify any underlying patterns.

This algebraic manipulation helps determine the subset relationships through congruence arguments. Proving whether one set is a subset of another involves checking if every element of one fits into the pattern established by the other. If all elements of set C can be expressed in the form 3s + 1, then C is a subset of D. Similarly, if elements of set D can be expressed as 6r - 5, then D is a subset of C. Exploring these subset questions touches upon important concepts of modular arithmetic and the relationships between linear combinations and integer sequences.

This requires breaking down the sequence definitions and investigating common integer properties to establish your proof or counterstatement. As you watch the solution, consider the logical steps required to convert these algebraic forms into concrete arguments about set inclusion. This problem elegantly illustrates the beauty of number theory as it applies to set theory, providing a doorway into deeper explorations of integer properties and sequences.