Chapter 4, Problem 5PS

(a)

To determine

To prove: The statement $\overline{z+w}=\overline{z}+\overline{w}$, where $z=a+bi$ and $w=c+di$.

(b)

To determine

To prove: The statement $\overline{z-w}=\overline{z}-\overline{w}$, where $z=a+bi$ and $w=c+di$.

(c)

To determine

To prove: The statement $\overline{zw}=\overline{z}\cdot \overline{w}$, where $z=a+bi$ and $w=c+di$.

(d)

To determine

To prove: The statement $\overline{\left(\frac{z}{w}\right)}=\frac{\overline{z}}{\overline{w}}$, where $z=a+bi$ and $w=c+di$.

(e)

To determine

To prove: The statement ${\left(\overline{z}\right)}^2=\overline{z^2}$, where $z=a+bi$.

(f)

To determine

To prove: The statement $\overline{\overline{z}}=z$, where $z=a+bi$.

(g)

To determine

To prove: The statement $\overline{z}=z\text{ when }z\text{ is real}$, where $z=a+bi$.