Question

Because there are infinitely many primes, we can assign each one a number: p0 = 2, p1 = 3, p2 = 5, and so forth. A finite multiset of naturals is like an ordinary finite set, except that an element can be included more than once and we care how many times it occurs. Two multisets are defined to be equal if they contain the same number of each natural. So {2, 4, 4, 5}, for example, is equal to {4, 2, 5, 4} but not to {4, 2, 2, 5}. We define a function f so that given any finite multiset S of naturals, f(S) is the product of a prime for each element of S. For example, f({2, 4, 4, 5} is p2p4p4p5 = 5 × 11 × 11 × 13 = 7865.
(a) Prove that f is a bijection from the set of all finite multisets of naturals to the set of positive naturals.
(b) The union of two multisets is taken by including all the elements of each, retaining du-plicates. For example, if S = {1, 2, 2, 5} and T = {0, 1, 1, 4}, S∪T = {0, 1, 1, 1, 2, 2, 4, 5}. How is f(S ∪ T) related to f(S) and f(T)?
(c) S is defined to be a submultiset of T if there is some multiset U such that S ∪U = T. If S ⊂ T, what can we say about f(S) and f(T)?
(d) The intersection of two multisets consists of the elements that occur in both, with each element occurring the same number of times as it does in the one where it occurs fewer times. For example, if S = {0, 1, 1, 2} and T = {0, 0, 1, 3}, S ∩ T = {0, 1}. How is f(S ∩ T) related to f(S) and f(T

Answer 1

NOTE:iI HOPE YOUR HAPPY WITH MY ANSWER....**PLEASE SUPPORT ME WITH YOUR RATING...THANK YOU.......