$\left|z-\left(1+i\right)\right|=1 \& \left|z-\left(1+i\right)\right|=2$ represents two concentric circles with centre $\left(1,1\right)$.

$\left|z-\left(1-i\right)\right|=1$ represents a circle with centre $\left(1,-1\right)$ and radius $1$

$A$ represents the region between the concentric circles with centre $\left(1,1\right)$

Set $B$ contains all points on minor arc $PQ$ of the circle $\left|z-\left(1-i\right)\right|=1$.

Hence, $B$ has infinite elements.