Explanation

(a)
We know that, $\left| \mathcal{- z} \right| = \left| \mathcal{z} \right|$
and $\left| \mathcal{z}_{1} + \mathcal{z}_{2} \right| \leq \left| \mathcal{z}_{1} \right| + \left| \mathcal{z}_{2} \right|$
Now, $\left| \mathcal{z} \right| + \left| \mathcal{z -}1 \right| = \left| \mathcal{z} \right| + \left| 1 - \mathcal{z} \right|$
$\geq \left| \mathcal{z +}\left( 1 - \mathcal{z} \right) \right| = |1| = 1$
Hence, minimum value of $\left| \mathcal{z} \right|\mathcal{+ |z -}1|$ is 1