Answer
a) $\{3\cdot a,3\cdot b,1\cdot c,4\cdot d\}$
b) $\{2\cdot a,2\cdot b\}$
c) $\{1\cdot a,1\cdot c\}$
d) $\{1\cdot b,4\cdot d\}$
e) $\{5\cdot a,5\cdot b,1\cdot c,4\cdot d\}$
Work Step by Step
We are given the multisets:
$$\begin{align*}
A&=\{3\cdot a,2\cdot b,1\cdot c\}\\
B&=\{2\cdot a,3\cdot b,4\cdot d\}.
\end{align*}$$
a) The union of two multisets is the multiset where the multiplicity of each element is the maximum of its multiplicities in the two multisets:
$$A\cup B=\{3\cdot a,3\cdot b,1\cdot c,4\cdot d\}.$$
b) The intersection of two multisets is the multiset where the multiplicity of each element is the minimum of its multiplicities in the two multisets:
$$A\cap B=\{2\cdot a,2\cdot b\}.$$
c) The difference $A-B$ of two multisets is the multiset where the multiplicity of each element is the multiplicity of the element in the first multiset $A$ less its multiplicity in the second set $B$; if this difference is negative, the multiplicity is $0$:
$$A-B=\{1\cdot a,1\cdot c\}.$$
d) In the same way as in part c) we calculate $B-A$:
$$B-A=\{1\cdot b,4\cdot d\}.$$
e) The sum $A+B$ of two multisets is the multiset where the multiplicity of each element is the sum of multiplicities in A and B.:
$$A+B=\{5\cdot a,5\cdot b,1\cdot c,4\cdot d\}.$$
