I'm interested in understanding the definition of set difference in multisets, but haven't been able to find a definition online.
For an example, suppose $A = \{a, b, b, c, c, c\}$ and $B = \{a, b, c\}$ are multisets. My question is, how is the set difference $A$ \ $B$ defined?
Is $A$ \ $B = \varnothing$, or the multiset $\{b, c, c\}$?
Using your example, we can write these multisets as functions, such as:
$A=\{a,b,b,c,c,c\}$ becomes $m_A (x) = [a]+2[b]+3[c]$
$B=\{a,b,c\}$ becomes $m_B (x) = [a]+[b]+[c]$
The multiset difference, $m_C (x)$, is defined as $m_C(x) = \text{max} \{m_A (x) - m_B (x),0\}$
$m_C (x)=[b]+2[c]$
$A\setminus B =\{b,c,c\}$
Note there are some consequences for this definition that may not align to the laws of classical set theory. See Singh etal (2016) for more details. Another reference that formalises multiset operations, such as set difference, is Syropoulos (2001).