Q:

Sketch the complex set

Sketch the set of complex numbers $\{z : |\bar z−1+i|<1\}$
Please note that the z in the inequality is CONJUGATED.
I did one where the z was not conjugated and ended up getting a circle and filling in the respective space. I am unsure of what a conjugate z does to getting the new graph.

A:

If your set is$$\left\{z\in\mathbb{C}\,\middle|\,\left|\overline z-1+i\right|<1\right\},$$then, since you know that $\left\{z\in\mathbb{C}\,\middle|\,\left|z-1+i\right|<1\right\}$ is the open disk centered at $1-i$ with radius $1$, your set is its conjugate, that is, it is the open disk centered at $1+i$ with radius $1$.

