Answer: I and III only

Deduce the seating.

• $A$ is in $X$ (given).
• "Only $E$ travels with $G$" → $E$ and $G$ share a car alone (just the two of them).
• $C$ is in $Z$.
• $B

otin Y$and$B

eq A$'s car, so$B$is in$Z$.

• $E, G$ cannot be in $Z$ (else $C$ would be with them, contradicting "only $E$ with $G$"). So $E, G$ are in $Y$.
• $X$ must hold $A$ + two others. The only people left are $D$ and $F$, so $X = \{A, D, F\}$.

Final assignment:

• $X = \{A, D, F\}$
• $Y = \{E, G\}$
• $Z = \{B, C\}$

Check the statements.

• I. No one travels alone. Smallest car has 2 people. ✓ True
• II. Only $D$ travels with $F$. $F$ is in $X$ with both $A$ and $D$, so $F$ has two companions. ✗ False
• III. Only $C$ travels with $B$. $Z = \{B, C\}$, so $B$'s sole companion is $C$. ✓ True