Today, we’ll examine the properties of the multiset intersection that we defined yesterday.
Suppose is a family of multisets over . Then the following relationships hold:
(ii) for all .
The proof of part (i) is a straightforward series of equivalencies:
Part (ii) follows directly from the definition of the infimum of a set, since for all and .
Recalling that the cardinal sum is a monotonic nondecreasing operation, we see that
for all , which proves part (iii).
Part (iv) requires only slightly more work. To begin with, consider a family of cardinals. If there exists some such that , then . If, however, no such exists, then
for all .
(This relies on the axiom of choice, which we have been assuming from the outset.) In other words, the product of any family of nonzero cardinals is monotonic nondecreasing.
Let . If there exists some such that , then the multiplicity of in the intersection would be 0, contradicting the fact that is a member of the intersection. Since for all , we have
For , we have
In either case, for all , and (iv) holds.
Copyright © 2008 Michael L. McCliment.