Multisets

A multiset is an unordered collection of elements where an element can occur as a member more than once. We can use the same notation for a multiset as we do for a set, but each element is listed the number of times it occurs.

So, the multiset denoted by $\left\{a,a,a,b,b\right\}$ is the multiset that contains the element $a$ thrice and $b$ twice. There is another notation $\left\{m_1\cdot a_1,m_2\cdot a_2,\dots ,m_r\cdot a_r\right\}$ denotes the multiset with $a_i$ occurs $m_i$ times.

The numbers $m_i$ are called the multiplicities of the elements $a_i$. (Elements not in a multiset are assigned $0$ as their multiplicities in this set)