We defined the union of a family of multisets last week. Today we’re going to look at some basic properties of the union operation on .
Suppose is a family of multisets over . Then the following relationships hold:
(ii) for all .
(iv) provided that all of the multisets have the same support.
The proof of part (i), like the similar result for multiset intersections, is a straightforward series of equivalencies:
Part (ii) follows directly from the definition of the supremum of a set, since for all and .
To prove part (iii), start by recalling that the sum of a family of cardinals is given by
where the sets are disjoint and for all . Therefore, letting be a family of disjoint sets such that , we have
for all and . This establishes that is an upper bound for in the linearly ordered set , and so . The result in part (iii) follows immediately since this holds for all .
The proof of part (iv) is similar, but relies on the definition of the product of a family of cardinals. The product of a family of cardinals is given by
where for all . Letting be a family of sets such that , we have
for all and , provided that for all . This establishes that is an upper bound for in the linearly ordered set whenever , which is the common support of all multisets in the family . If is not in this support, then . In either case, , and part (iv) follows immediately since this holds for all .
The requirement that all multisets in the family have the same support is necessary; you can see this by considering the product and union of the multisets and .
Finally, part (v) follows immediately from the fact that
for all and .
Copyright © 2008 Michael L. McCliment.