Answer
The Venn diagram is as follows:
Work Step by Step
First, we find the four intersections of the provided sets:
\[\begin{align}
& A\cap B=\left\{ {{x}_{3}},\ {{x}_{9}} \right\}\cap \left\{ {{x}_{1}},\ {{x}_{2}},\ {{x}_{3}},\ {{x}_{5}},\ {{x}_{6}} \right\} \\
& =\left\{ {{x}_{3}} \right\}
\end{align}\]
\[\begin{align}
& B\cap C=\left\{ {{x}_{1}},\ {{x}_{2}},\ {{x}_{3}},\ {{x}_{5}},\ {{x}_{6}} \right\}\cap \left\{ {{x}_{3}},\ {{x}_{4}},\ {{x}_{5}},\ {{x}_{6}},\ {{x}_{9}} \right\} \\
& =\left\{ {{x}_{3}},\ {{x}_{5}},\ {{x}_{6}} \right\}
\end{align}\]
\[\begin{align}
& A\cap C=\left\{ {{x}_{3}},\ {{x}_{9}} \right\}\cap \left\{ {{x}_{3}},\ {{x}_{4}},\ {{x}_{5}},\ {{x}_{6}},\ {{x}_{9}} \right\} \\
& =\left\{ {{x}_{3}},\ {{x}_{9}} \right\}
\end{align}\]
\[\begin{align}
& A\cap B\cap C=\left\{ {{x}_{3}},\ {{x}_{9}} \right\}\cap \left\{ {{x}_{1}},\ {{x}_{2}},\ {{x}_{3}},\ {{x}_{5}},\ {{x}_{6}} \right\}\cap \left\{ {{x}_{3}},\ {{x}_{4}},\ {{x}_{5}},\ {{x}_{6}},\ {{x}_{9}} \right\} \\
& =\left\{ {{x}_{3}} \right\}
\end{align}\]
Now, place the elements in the regions formed by the above intersections:
First, place the elements of \[A\cap B\cap C=\left\{ {{x}_{3}} \right\}\] in the innermost region V.
Then, place those elements of \[A\cap B\]in the region II, which do not belong to the region V, which are 5 and 6.
Place the elements of \[A\cap C\]in the region IV, which donot belong to the region V, this region is empty since 4 already came in region V.
Then, place the elements of \[B\cap C\]in the region VI, which do not belong to the region V. So, it contains 7 only.
Then, put the remaining elements of sets A, B, and C in the regionsI, III, and VII, respectively.
So region I contains 8, region III contains 1 and 2, and region VII contains 3 only. Finally, region VIII contains the remaining elements of the set U. So, all the elements of the set U are used up. So, it contains only 9.
