Among my favorite Youtube channels is that of Prof N J Wildberger, on which he uploads explanatory videos on different topics in mathematics (along with lectures he gave in UNSW). The latest video is on multisets, and it happens to be related to what I am currently working with. So I figured I would compose a short proof for the interesting equivalence he pointed out at the 11th minute mark.

Theorem:

For any multisets A and B

Proof:

Firstly, let $a_k$ denote some element in $A$ or $B$ and let’s establish that

Now let $n_i$ denote the number of $a_i$ occurring in $(A \cup B)$,
$p_i$ denotes the number of $a_i$ occurring in $(A \cap B)$, and
$v_i$ to denotes the number of $a_i$ occurring in $(A + B)$.

The equivalence theorem above can be rewritten as:

According to the definitions of union and intersection between multisets:

we can further rewrite the formula as

and that is equivalent as stating

which is trivially true.