Answer
$(D'\cap U)\cup E=\{ 3,4,5,6,7,8,9,10 \}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To get the elements of $
(D'\cap U)\cup E
,$ combine the elements of $D'$ with $E$.
$\bf{\text{Solution Details:}}$
The elements of the complement of $D$, given by $D'$ are all the elements of $U$ that are not in $D$. With $D=\{1,2,3 \}$ and $U=\{ 1,2,3,...,9,10 \}$, then
\begin{array}{l}\require{cancel}
D'=\{ 4,5,6,...,9,10 \}
.\end{array}
The intersection of $D'$ and $U,$ given by $D'\cap U$, are all the common elements of $D'$ and $U$. These common elements are also the elements of $D'.$ Hence,
\begin{array}{l}\require{cancel}
D'\cap U=D'
.\end{array}
Adding the elements of $E=\{ 3,7 \}$, then
\begin{array}{l}\require{cancel}
D'\cup E=\{ 3,4,5,...,9,10 \}
.\end{array}
THerefore, $
(D'\cap U)\cup E=\{ 3,4,5,6,7,8,9,10 \}
.$
