14. There are three interest groups $A$, $B$, and $C$, with 5, 4, and 7 members respectively. Among them, 3 members are in both $A$ and $B$ interest groups, and 2 members are in $A$, $B$, and $C$ interest groups. Let $(A \cup B) \cap C$ consist of $x$ members, and $A \cup B \cup C$ consist of $y$ members. Find the range of values for $x$ and $y$.
14. Let $A$, $B$, and $C$ represent the sets of members of the respective groups. From the given information, we have $|A|=5$, $|B|=4$, $|C|=7$, $|A \cap B|=3$, and $|A \cap B \cap C|=2$.
Since $|A \cap B \cap C|=2$, it follows that $|(A \cup B) \cap C| \geqslant 2$.
Since $|A \cap B|=3$ and $|A \cap B \cap C|=2$, there exists an element $a \in A$ and $a \in B$, but $a \notin C$. Therefore, there exists an element $a \in A \cup B$, but $a \notin C$. Also, $|A \cup B|=|A|+|B|-|A \cap B|=5+4-3=6$, so $|(A \cup B) \cap C| \leqslant 5$. $|A \cup B \cup C|=|A|+|B|+|C|-|A \cap B|-|B \cap C|-|C \cap A|+|A \cap B \cap C|=5+4+7-3-|B \cap C|-|C \cap A|+2=15-|B \cap C|-|C \cap A|$.
Since $|A \cap B \cap C|=2$, we have $|B \cap C| \geqslant 2$ and $|C \cap A| \geqslant 2$, so $|B \cap C|+|C \cap A| \geqslant 4$. Substituting this into (2) gives $|A \cup B \cup C| \leqslant 11$. From (1) and $|A|=5$, $|B|=4$, $|C|=7$, we have $|B \cap C| \leqslant 3$ and $|C \cap A| \leqslant 4$, so $|B \cap C|+|C \cap A| \leqslant 7$. Substituting this into (2) gives $|A \cup B \cup C| \geqslant 8$.

Examples where the equality holds: $A=\{1,2,3,4,5\}$, $B=\{1,2,3,6\}$, $C=\{2,3,7,8,9,10,11\}$, in this case $|(A \cup B) \cap C|=2$ and $|(A \cup B) \cup C|=11$; $A=\{1,2,3,4,5\}$, $B=\{1,2,3,6\}$, $C=\{2,3,4,5,6,7,8\}$, in this case $|(A \cup B) \cap C|=5$ and $|A \cup B \cup C|=8$. Therefore, $2 \leqslant x \leqslant 5$, $8 \leqslant y \leqslant 11$, $x, y \in \mathbf{N}_{+}$.