1. Given sets:
$$
\begin{array}{l}
A=\left\{x \mid x=2 n-1, n \in \mathbf{Z}_{+}\right\}, \\
B=\{y \mid y=3 x-1, x \in A\} .
\end{array}
$$

Then $A \cap B=$
$-1 . \varnothing$.
Since the elements in set $A$ are positive odd numbers, and the elements in set $B$ are even numbers, therefore, $A \cap B=\varnothing$.