  We construct a minimal complex surface of general type with $p_g=0$, $K^2
=4$, and $\pi_1=\mathbb{Z}/2\mathbb{Z}$ using a rational blow-down surgery and
a $\mathbb{Q}$-Gorenstein smoothing theory. In a similar fashion, we also
construct a symplectic 4-manifold with $b_2^+=1$, $K^2=5$, and
$\pi_1=\mathbb{Z}/2\mathbb{Z}$.
