Set theory: Problem 4


Do you understand the set-builder definition of L(s,t)? L(s,t) consists of all integers that can be expressed as the sum of multiples of s and t. If you can't form a picture of what this might mean, pick some concrete values for s and t. Calculate some values that live in L(s,t). Suppose s=6 and t=9. Find some integers that cannot be written in the form sx+ty (where you get to pick the integers x and y).

You need to prove \(L(a,m) \subseteq L(r,m)\). So pick an element x in L(a,m). You need to show that x is in L(r,m). Remember that L(a,m) and L(r,m) contain integers, so x will be an integer.

You've been asked to use the textbook definitions of congruence and divisibility. Look them up if you don't remember which form was our official definition.