Properties of multiset intersection

Today, we’ll examine the properties of the multiset intersection that we defined yesterday.

Suppose \left\{\mathcal{M}_i = \left(X, f_i\right)\right\}_{i\in I} is a family of multisets over X. Then the following relationships hold:

(i) \mathrm{support}\left(\bigcap_{i\in I}{\mathcal{M}_i}\right) = \bigcap_{i\in I}{\mathrm{support}\left(\mathcal{M}_i\right)}.

(ii) \bigcap_{i\in I}{\mathcal{M}_i} \subseteq \mathcal{M}_i for all i\in I.

(iii) \bigcap_{i\in I}{\mathcal{M}_i} \subseteq \biguplus_{i\in I}{\mathcal{M}_i}.

(iv) \bigcap_{i\in I}{\mathcal{M}_i} \subseteq \:\cdot\!\!\!\!\!\!\;\bigcup_{i\in I}{\mathcal{M}_i}.

The proof of part (i) is a straightforward series of equivalencies:

\begin{array}{r@{\:\Leftrightarrow\:}l} x\in \mathrm{support}\left(\bigcap_{i\in I}{\mathcal{M}_i}\right) & \inf f_i(x) \neq 0 \\ & \left(\forall i\in I\right)\: f_i\left(x\right)\neq 0 \\ & \left(\forall i\in I\right)\: x\in\mathrm{support}\left(\mathcal{M}_i\right) \\ & x\in\bigcap_{i\in I}{\mathrm{support}\left(\mathcal{M}_i\right)}. \end{array}

Part (ii) follows directly from the definition of the infimum of a set, since \inf f_i\left(x\right) \leq f_i\left(x\right) for all x\in X and i\in I.

Recalling that the cardinal sum is a monotonic nondecreasing operation, we see that

\inf f_i\left(x\right) \leq f_i\left(x\right) \leq \sum_{i\in I}{f_i\left(x\right)}

for all x\in X, which proves part (iii).

Part (iv) requires only slightly more work. To begin with, consider a family \left\{\kappa_i\right\}_{i\in I} of cardinals. If there exists some i\in I such that \kappa_i = 0, then \prod_{i\in I}{\kappa_i} = 0. If, however, no such i\in I exists, then

\kappa_i\leq\prod_{i\in I}{\kappa_i} for all i\in I.

(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 x\in\bigcap_{i\in I}{\mathcal{M}_i}. If there exists some i\in I such that x\not\in\mathcal{M}_i, then the multiplicity of x in the intersection would be 0, contradicting the fact that x is a member of the intersection. Since f_i\left(x\right)\neq 0 for all i\in I, we have

\inf f_i\left(x\right) \leq f_i\left(x\right) \leq \prod_{i\in I}{f_i\left(x\right)}.

For x\not\in\bigcap_{i\in I}{\mathcal{M}_i}, we have

\inf f_i\left(x\right) = 0 = \prod_{i\in I}{f_i\left(x\right)}.

In either case, \inf f_i\left(x\right) \leq \prod_{i\in I}{f_i\left(x\right)} for all x\in X, and (iv) holds.

Copyright © 2008 Michael L. McCliment.


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